Seeing Solutions: Using Slope Fields for Differential Equations
TLDRIn this engaging calculus session, Steve and Tom explore differential equations and slope fields using Texas Instruments technology. They delve into various equations, demonstrating how to connect equations with corresponding slope fields and solve initial value problems. The session highlights the utility of technology in visualizing and understanding complex mathematical concepts, emphasizing the importance of these topics for both teachers and students.
Takeaways
- π The session focuses on differential equations and smoke fields, which are important topics for both teachers and students preparing for the AP Calculus exam.
- π The TI-Inspire and TI-84 calculators are used to demonstrate the plotting of slope fields and solution curves for differential equations.
- π Slope fields are visual representations that help in understanding the behavior of solutions to differential equations and can be plotted using technology.
- π― When creating a slope field, the expression for the slope of the differential equation is evaluated at various points in the plane to determine the direction of the solution curves.
- π It's important to note that while slope fields can be generated using technology, students are expected to understand and be able to sketch slope fields without calculators in the AP Calculus exam.
- π€ The video script includes a discussion on how to handle the constant of integration (C) when solving differential equations and how it may change form during the process.
- π The script provides examples of solving differential equations by separating variables and integrating both sides, highlighting the steps and considerations involved in the process.
- π The session also addresses the concept of domain, emphasizing the need to consider the domain when working with functions and their graphs, especially in relation to the constant of integration.
- π‘ The use of technology, such as Mathematica, is mentioned for enhancing the visualization of slope fields and solution curves, providing additional insights into the behavior of solutions.
- π The session encourages participants to think critically about the relationships between differential equations, their slope fields, and the resulting solution curves.
- π The transcript ends with a discussion on the use of vector direction fields for applications such as golf, showing the practical applications of mathematical concepts.
Q & A
What is the main topic of discussion in the transcript?
-The main topic of discussion in the transcript is differential equations and smoke fields, specifically focusing on how to use the TI Inspire and TI-84 calculators to plot and analyze them.
What is a slope field plot?
-A slope field plot is a graphical representation of the slopes of a differential equation at various points in the coordinate plane. It is used to visualize the behavior of solutions to the differential equation.
How does the speaker describe the process of plotting a slope field on the TI Inspire?
-The speaker describes the process of plotting a slope field on the TI Inspire by first opening a graph page, selecting 'graph entry edit' from the menu, choosing 'differential equations', and then entering the differential equation into the expression field. The slope field is generated by evaluating the slope expression at each point in the visible window.
What is the significance of the expression x^2 / 4 in the context of the slope field plot?
-The expression x^2 / 4 represents the slope of the differential equation at any point in the coordinate plane. It is used as an example to demonstrate how the slope field plot can differ from the expected graph of the function, in this case, a parabola.
What is the purpose of the 'field resolution' setting in the TI Inspire?
-The 'field resolution' setting controls the number of line segments or the fineness of the lattice of points plotted for the slope field. Increasing the field resolution results in a more detailed and smoother slope field plot.
How does the speaker address the use of slope field plots in the context of the AP Calculus exam?
-The speaker mentions that slope field questions are common on the AP Calculus exam, usually in the free response portion. However, they note that while technology like the TI Inspire can be used for practice and to check the plausibility of solutions, the actual exam expects students to sketch the slope field without the aid of technology.
What is the differential equation given in the example where the speaker discusses the slope being zero along a line?
-The differential equation given in the example is dy/dx = y - x/2.
What is the initial condition used in the problem where the speaker finds a C value of 8/3?
-The initial condition used in that problem is y(1) = 4.
How does the speaker justify the absence of the slope field in quadrants 2 and 4 for a particular differential equation?
-The speaker suggests that the absence of the slope field in quadrants 2 and 4 is due to the square root of the product of x and y, which would be undefined for negative values in those quadrants.
What is the final expression for y derived by the speaker for the differential equation dy/dx = x + 1 / (x*y)?
-The final expression for y derived by the speaker is y = (3^(8/3) * x^(4/3)) / (1 + (3^(8/3) * x^(4/3))^2).
What does the speaker mention about the use of technology in solving differential equations?
-The speaker mentions that while technology, such as the TI-84 and TI Inspire calculators, can be very helpful in visualizing and solving differential equations, students are expected to understand the concepts and be able to solve problems without technology during exams.
Outlines
π Introduction to Differential Equations and Smoke Fields
The paragraph introduces the topic of differential equations and smoke fields, highlighting their significance in the context of AP Calculus exams. The discussion involves Steve Kokaska and Tom Dick, who are preparing to answer questions related to this topic. The introduction also touches on the use of Texas Instruments equipment for demonstrating the solution of differential equations, with a focus on the TI-Inspire calculator's capabilities.
π Exploring the TI-Inspire Calculator for Differential Equations
This section delves into the specifics of using the TI-Inspire calculator to plot and analyze differential equations. The conversation explains how to navigate the calculator's menu to access the differential equation plotter and how to input equations to generate slope fields. It also discusses the importance of understanding the relationship between the differential equation and the resulting slope field, emphasizing the educational value of visual representation in grasping the concept.
π Analyzing the Characteristics of Slope Fields
The paragraph focuses on the characteristics of slope fields and how they relate to the original differential equations. It discusses the expectations students might have when visualizing slope fields and the common surprises they encounter. The explanation includes a detailed look at how the sign and magnitude of the derivative affect the slope field's appearance, using the example of x^2/4 to illustrate the concept.
π Addressing Exam Expectations and Technology Use
This part of the discussion addresses the expectations for slope field questions on exams, emphasizing that while technology can aid in understanding, students will need to sketch slope fields without calculator support during the exam. It also touches on the availability of slope field programs for the TI-84 calculator and the importance of practice in using this tool for both educational and exam purposes.
π€ Reflecting on the Process of Matching Differential Equations to Slope Fields
The paragraph involves a step-by-step reflection on how to match given differential equations to their corresponding slope fields. It outlines strategies for analyzing the equations, such as considering constant values, examining the first quadrant, and identifying special cases within the equation. The discussion also includes an interactive element, with the presenter solving problems and engaging the audience with questions.
π Solving Differential Equations and Discussing Constants
This section focuses on the process of solving differential equations, with an emphasis on the importance of correctly handling constants. The explanation involves separating variables, integrating both sides of the equation, and using initial conditions to find the specific solution. The paragraph also includes a practical demonstration of solving a differential equation and the considerations involved in isolating the variable y.
π Examining the Relationship Between Equations and Slope Fields
The paragraph explores the relationship between the solutions of differential equations and their corresponding slope fields. It involves a detailed analysis of how the solutions fit within the slope fields, using specific examples to illustrate the points. The discussion also raises open questions for further exploration, encouraging engagement with the material and promoting a deeper understanding of the concepts.
π Investigating the Impact of Variables on Slope Fields
This part of the discussion investigates how changes in variables affect the slope fields. It involves a detailed examination of the effects of increasing or decreasing variables on the slope of the field, and how this relates to the original differential equations. The explanation includes visual aids and interactive elements to enhance understanding and retention of the material.
π€ Collaborative Problem-Solving and Technology Integration
The paragraph showcases a collaborative approach to solving differential equations, with a focus on integrating technology into the process. It involves a back-and-forth exchange between the presenters as they work through problems, use technology to visualize solutions, and discuss the implications of their findings. The conversation also includes audience engagement, with questions and suggestions being incorporated into the discussion.
π Final Thoughts and Open Questions
The conclusion of the discussion involves a wrap-up of the main points covered, along with a reflection on the use of technology in teaching and understanding differential equations. It also includes a mention of open questions raised during the session, encouraging further exploration and discussion. The paragraph ends with a note on the educational value of the content and the potential for future engagement.
Mindmap
Keywords
π‘Differential Equations
π‘Slope Field
π‘Texas Instruments Calculators
π‘AP Calculus Exam
π‘Smoke Fields
π‘Logarithms
π‘Integration
π‘Initial Value Problems
π‘Technology in Education
π‘Mathematica
Highlights
Introduction to differential equations and smoke fields as a topic of interest for the AP Calculus exam.
Use of TI Inspire and TI-84 calculators for exploring differential equations and slope field plots.
Explanation of the slope field plot as a visual representation of the derivative of a function.
Demonstration of how to enter a differential equation into the calculator and generate a slope field plot.
Discussion on the importance of understanding the relationship between the differential equation and the corresponding slope field.
Illustration of how the slope field can help verify the solution of a differential equation.
Explanation of the process for solving separable differential equations and the inclusion of the constant term.
Use of technology to sketch slope fields and solution curves for differential equations.
Discussion on the limitations and capabilities of the calculator when it comes to graphing in different quadrants.
Application of initial value problems in differential equations and how to use them to find specific solutions.
Demonstration of how to use the initial condition to find the value of the constant in a differential equation.
Explanation of the relationship between the domain of a function and its representation on a slope field.
Discussion on the use of vector direction fields for visualizing multiple solution curves.
Explanation of how to interpret the behavior of a solution curve in relation to the slope field.
Demonstration of the process for solving a differential equation involving the log of an absolute value.
Discussion on the importance of understanding the domain and range of functions when working with differential equations.
Conclusion and wrap up of the key points discussed during the session, including the practical applications of slope fields in understanding differential equations.
Transcripts
well good evening and welcome to another
edition of Monday Night calculus my name
is Steve kokaska and I'm joined this
evening by my colleague Tom Dick from
Oregon State and in the background is
our Texas Instruments expert Allison
steel uh Curtis could not make it
tonight but Allison will be monitoring
than chat as usual and we may be able to
ask her a few questions I'm not sure um
tonight's topic has to do with
differential equations and smoke Fields
this is a really enjoyable topic for me
we always get lots of good questions
about this concerning this topic and we
usually see a question on differential
equations on the AP Calculus exam
usually on the free response portion so
I think this is a very important topic
for teachers and students um we've
actually had some good questions uh sent
to Curtis prior to tonight and we'll try
to get to answer all of those and Tom
will make sure uh that I don't miss any
of those so once again Allison will be
monitoring the chat if you have any
questions and if you have any answers to
our questions uh please do correspond
with her and we're going to start
tonight with a little technology Tom's
going to look at I think the ti Inspire
first is that
correct uh yes Steve I'll take a look at
both the SL fields on the uh the Inspire
and on the 84 so okay all yours
Tommy
okay all right I'm going to start out by
uh sharing a TI Inspire screen
okay and let's see if that's uh come up
there I think it's coming up
okay and all right uh let me uh let's do
a little skape there okay so what I've
done is uh open this uh uh TI inspired
document up in a in a graphs page and
this is uh probably the graphs uh
environment that most of us would use
probably 80 90% of the time would be the
function environment uh but what I'm
going to do is go over to the
menu and uh let's go down to three graph
entry edit that's going to have our uh
different kinds of graphs that we can
look at and you can see there we are on
number one is function and what I want
to do is scroll down all the
way to this thing that says oops I'm
going to hit up Arrow it should wrap
back around there we go def differential
equations that's the plotter uh but
people should keep in mind that that is
where you're going to find What's called
the the slope field plot and that's
something that's been in the course
description now for a couple of decades
but it's a uh it was pretty radical when
it first came into the the course yes it
was
was yeah so let me go ahead and hit that
and here we see a lot different kind of
Entry screen here uh for entering our
differential equation and it's kind of
spaced a little bit funny but you you
can see a y1 there but you see way over
here this Prime so this is actually the
derivative of y1 and now we can put it
in an expression
that's going to describe the
slope of
y1 at any point and so this expression
could involve both X and Y uh I'm going
to start out just by putting in an
expression with X so let me try x
squared and I'm going to divide that
that by uh four okay so I've got x^2 / 4
and uh down below the
expression you can see there is a space
for you to put in an initial condition
that would be a single point that you
would like to be make sure the solution
curve goes through I'm not going to make
use to that right now I'm just going to
make use of this and so when I um hit
graph on this
so we enter this we get what's called a
slope field now all a slope field is is
there's a a lattice or sampling of
points throughout the plane here in the
visible window and at each of the points
here it's used that expression to
evaluate a slope and then it plots a
little segment having that slope now
whenever I've introduced slope fields to
students for the first time I often use
an example like this something that they
could very
easily uh figure out what the original
y1 form would look
like uh but they're often very surprised
when they see this slope field because
when they think of x^2 over 4 the image
that comes to their mind immediately is
a parabola and so they're expecting a
parabola to fit the slope field but
that's not really the case and we can
think about it let's see x^2 over
4 uh that is its value is always going
to be positive except at zero it would
be zero but it will never have a
negative value that means all of the
little line segments we see in the slope
field should have a positive slope and
we can see when X is close to zero well
x^2 over 4 would be very small and so
we'll have a very small positive slope
as X gets larger in either the positive
direction or large magnitude negative
Direction x^2 over4 will be a larger
number and now we're seeing a steep
positive slope so this is a slope field
that's
fitting um those slope values that were
generated there now I'm going to go
ahead and uh uh go back to the entry
thing and and show you a setting that
you may want to take advantage of notice
here it's uh kind of defaulted back for
a second differential equation I'm going
to hit the up Arrow to get back to my
original
here and what I'm going to do is Arrow
over and let's see I'm hoping that's um
what I'm trying okay I got into this
position you see this dot dot dot to the
the right there that's where I want to
go and that dot dot dot is just a Shand
has there's a lot of settings in that
dot dot dot that we can take advantage
of so I'm going to hit
enter and I get this long scroll down
menu uh you can see there's a you can
choose like Oilers method for plotting
your curves which is is a different
topic that we're not going to touch on
tonight but let me just scroll down and
show you the one setting that I changed
very often when I'm working with the
Inspire and that's the field resolution
and the field resolution is just
controlling how many line segments or
how fine of lattice of points is plotted
for your slope field and I find 14 is a
little bit coarse it's a little bit
rough and so what I tend to do is I
usually just double that so instead of
14 I'm going to put in
[Music]
28 and then I'll say
okay and let's see let me
uh escape to get the
uh um entry line off of there and now
you're seeing a slope field for the
exact same differential equation but
it's much finer and also I think now you
can see like I said x^2 over 4 what
function would have that as a derivative
well it's going to be what a cubic
actually X cubed over 12 I think if I
did my arithmetic right but you can see
that a cubic would fit the slope field
very nicely in fact we can try that out
by going back to the
menu go back to graph entry
edit and now get the function on there
and without erasing the slope
field we could actually enter a function
that we think might fit the slope field
so I'm going to go ahead and see if
I if my uh calculation in my head was
right I'm going to put in
x
cubed and I'm going to divide
that by 12 whoops I think I might have
accidentally entered didn't I okay so
let
me get back
up sorry about that okay let me get back
in the entry I wanted to
divide by 12 mhm by the way when I look
at the what I did plot I can tell that's
not going to work plain old X cubed is
clearly not it's actually crossing a lot
of these line segments at an angle
that's just not fitting so let me amend
that put in X cubed over
12 and now I'm seeing a much better fit
to my SL field so that's just kind of
the basics of smoke field and and how
you can plot over it
uh there is also a uh this this is um a
functionality that's resident on the
Inspire does you don't have to have cast
or anything it's it's on all the
inspires uh however on the
ti84 this is the one spot where if
there's one program that you load onto
your 84 for calculus I think this would
be the one you'd want to program for
slope field and I don't know if Steve
mentioned this but slope field questions
uh on the
exam uh are almost always going to be on
the non-calculator part of the exam so
when those are assessed they're going to
for example expect you maybe to sketch a
slope field on your own but there in
that setting it wouldn't make sense for
people to have access to something that
would just generate the slope wheel for
you right uh nevertheless they're a
fantastic tool they're a great thing to
have technology generated but when
they're testing whether you understand
what a slope field is they're going to
do that on the non- calculated part of
the exam so I'm going to do a uh just a
quick stop share and then share back
again Steve if that's okay yeah sure and
switch over to the
ti84 and let's see if I'm able to bring
that up okay okay here we go there's the
ti84 and on the ti84 uh
there's not a a resident slope field
program uh but there is one that you can
download for free off of the ti
education.com
site um and we'll make sure people have
some information we'll communicate that
to Curtis and Allison so we can make
sure people know how to get that if I go
to my programs here this ti4 already has
one in
there and it's uh just named SLP FLD
short hand for slope field so that's the
program and the way this slope field
program works is I'm going to go to y
equals and it's going to expect to see
the differential equation for dydx
stored in
y1 now that means if you wanted to put
in something like an expression in terms
of X and Y let's do one like that say
negative
x
/
y that's going to look really peculiar
to you because you never see why in the
y equals menu as in there but keep in
mind that this is just an
expression and y1 if I have any
numerical value for x and y stored y1
will
evaluate using both of those values okay
uh so this is where we're storing our
differential equation so dydx is equal
tox over
Y
and if I now quit out of
there uh I can go to
programs and there's my slope field I
want to execute that
program
and let's see I've got a little bit
different window than a St standard one
but it's good enough okay so here's a
whatever window I had it noticed it's
doing a nice slope feel for me just like
on the Inspire but it was a program that
I had to load on there uh this might
look like a ripples of a you know like
throwing a stone into a a pond and
seeing the ripples come out um that's
actually not by accident at all if you
think about that Negative X over y
that's that's a derivative you may have
encountered when you doing implicit
differentiation if you do implicit
differentiation on a circle equation
like an X2 + y^2 = 9 or something like
that the implicit derivative turns out
to Bex over y so we can see the solution
Curves in this case are going to be
parts of circles so that's kind of cool
and again uh if you did want to plot a
function on top of
this you could just use use Y2 or Y3 or
any any of the other uh
equations y equals things notice the the
soap Field's already on there it's not
selected but I'll go ahead in Y2 I'm
going to put in
the square
root of I mentioned x^2 + y^2 = 9 well
let me solve that for y so that y would
be the square root of the quantity 9 -
x^2
and let me graph
that and boom there it is there's a nice
curve and it's following the fit of the
slope field so um again uh slope Fields
uh on the exam it's probably going to be
on the non- calculated part of the exam
but for practice uh and and for
especially for seeing whether the
solution to a differential equation is
uh plausible what I mean makes sense is
you could do a slope field for it and
then take your solution and plot it on
top of the slope field and see if it
fits if you see any uh discrepancies
that's a Surefire tip that something's
gone ay in your calculation somewhere
that you need to take a look at all
right I think I've wagged my tongue
enough Steve I'm G to turn back over to
you and so you can dive into your
questions so while
we're go ahead while we're doing that I
did just want to give a shout out we've
got quite a few joining us from Miss
Connelly's class as well as Mr uh the
famous Mr Crowley's class this evening
all right
great shout out to Mark hey Tom before
before you leave the screen yes sir one
quick question uh so what do you do in
order to sort of erase the slope
field is there some special command for
that so now all done with this and I
want to graph another function I'm going
on to another problem how do I sort of
erase this and start over yeah no no
good point uh actually you know any
change at all you would make to the
window
would get rid of the slope field so for
example if I was to just do a
zoom decimal here
yes I see okay I've got a fresh screen
it went ahead and still graphed my that
function I had entered into Y2 okay but
slope field for me to generate a new
slope field in a new window I also need
to rerun the program gotcha so if I
change if I say oh the slope field I
really want to look at it at a different
part or a different window I'd go change
my window but then I would need to go to
quit to the kind of the home calculation
screen pull up the program just run it
again gotcha okay yeah no that's
excellent question is if you're wanting
to wipe the Slate clean you know just
change the window and then you could
even go back to the old window you had
but now you won't have the slope field
there anymore okay all right okay very
good all right I'll stop the share it's
all your Steve all right well let's take
a look if we can here at some of the
problems and
solutions that we posted on differential
equations and slope Fields so I'm going
to try to write a little bit here here
and this first problem involves trying
to connect uh the differential equation
with the corresponding slope field and I
think Tom correct me if I'm wrong but
this is the kind of a problem that one
might see a student might see in the
multiple choice portion of the exam
might abely might get a differential
equation and say okay which of these
four slope Wheels corresponds to this te
okay so when I solve these problems I
mean I'm always looking for some sort of
prescriptive way to do this kind of a
stepbystep process and I don't know if
there is one but here are a couple of
things that I do in trying to figure
this out um I might look at what happens
when I hold say y
constant and I might look along for
example a horizontal line and I might
ask well does the does the slope change
do those slopes of the line segments
change as X
varies well if they don't well that
would suggest that X is not involved in
the differential
equation and similarly I might look
along a line where X is constant and see
well are those changing as y varies if
they're
not then that suggests to me that y
might not be
involved now another thing that I
frequently do is I take a look at the
first quadrant where both X and Y are
positive that somehow seems to be the
easiest the most intuitive for all of us
and it's almost easy to plug in a couple
of values and see what's going on so I
always look in the first
quadrant and finally the last thing that
I do is I kind of look for any special
circumstances in the differential
equation for example are there any
places where I know that the derivative
is going to be zero and can I can I
identify that pattern in the slope
field so I'm actually going to start
there by looking at
a if I look at that differential
equation one of the first things I
thought about is H when will that
derivative be zero well let's see if I
can scribble a little bit up here Tom
let me know if you can see this if I
scribble a little bit up here if y were
equal to X
over2 then that derivative would be
zero and yal X over2 is a straight line
through the
origin so if I can find any one of these
slope Fields where in fact I have zero
slope along that line yal over2 I'm
feeling pretty good about this and in
fact if I look real closely over here
I'm going to scribble a little to I
think there is a line in there there it
is YX
over2 where in fact the slope is
zero now I can also argue now that I've
got that I can also argue think about
what's going on above that line is y
bigger than x over2 is it less than x
over2 and how does that affect the slope
and take a look at those line segments
but boy that's a very good start on that
one so a corresponds to slope field
two we going to look at
B here's what I saw over here in B I saw
a product and I thought well how do I
make that product zero the only way to
do that is if one of the factors is zero
so that means if x is0 or y is equal to
-2 the derivative is going to be zero so
I'm looking down here at slope field
three right here where Y is equal to
two I have those line segments with
slope zero as as expected in that
differential equation and right here
sorry about this kind of funny Arrow but
right there along the Y AIS where X is
equal to zero if you can see those
little GRE green line segments they have
slope zero also that's pretty
cool now I think another thing you can
see here is probably in the first
quadrant pretty easily where both X and
Y are positive in fact where X is
positive and Y is greater than minus two
the product there is going to be
positive and the slope going to be
positive and that's over in this area
here and I'm feeling pretty good about
this I think B is connected with slope
field three
cool see how about this one well here's
what I saw over here in this one once
again I was thinking about well when is
that derivative going to be zero well I
I can kind of see that equation on the
right hand side and I'm thinking you
know that's kind of related to a circle
I'm going to scribble a little bit off
to the side here x^2 + y^2 = 4 well I
know that if I pick an X and A Y
that satisfy that equation it's on that
Circle and that's going to make that
derivative
zero and so it seems to me that on this
circle x^2 + y^2 = 4 man those line
segments better have slope zero now I
didn't take a very fine resolution as
Tom was talking about but I think you
can kind of see along this
circle the line segments have slope
zero what happens outside that Circle
well outside that
Circle x^2 + y^2 is going to be greater
than four and therefore the derivative
is positive and so look at all those
line segments outside that Circle they
all have positive slope I'm feeling good
about this one and inside the circle
that expression x^2 + y^2 is less than 4
and so I better have some line segments
that are negative and in fact they are
which is pretty
cool so C goes with
four ah
finally this is do this one Steve I
think I can do this one what's that I
think I can do this one you can you can
match this
one well this is kind of a cool one
maybe I should have done this one first
y Prime is the cosine e to Theus XY I
I'm not sure exactly how to start on
this one except maybe looking in the
first
quadrant I mean what happens when both X
and Y are positive and what happens when
both X and Y increase get
large well let's see when both X and Y
are large that product is large and this
is very large negative what does that
mean for E to that value well that makes
e to that expression e to that value
small close to
Z and what's the cosine of zero a value
close to zero son of a
gun that's going to give me a
one and so it seems as X and Y get large
increase without
bound that argument of the cosine is
getting very close to zero and so the
derivative should be getting close to
one and in fact that's what's happening
over here and that's exactly what
happens over here because say in the
third
quadrant X and Y have the same
sign so their product is positive and
then there's that extra negative there
in the
exponent which once again drives that
argument close to
zero and we get a slope of
one that's really cool over here in the
second and the fourth quadrant I think
if you consider say look along a
horizontal lines
and let X vary let X increase I think
you can argue that there's going to be
some sort of cycle here we know that the
cosine cycles and there's going to be
some sort of cycle here uh as X
increases it's not quite evident in the
second and the fourth quadrant in my
drawing again my resolution isn't that
great but I'm pretty sure d goes with
one Co Steve yes sir um like on your
part A there
on the exam will they include that Roman
numeral too right after
it I don't think so Tom okay we could
Lobby for
[Laughter]
that good
question all right I have a couple of
problems here in in two where we're just
going to solve the differential equation
these are not initial value problems
we're simply going to solve the initial
solve the differential equation um a
couple of the questions that was sent to
Curtis dealt with including this
constant so I'm going to try to make
sure that I'm very careful as I speak
about these constants and when they
arrive when they should be put into this
equation when they should
appear all right so the only kind of
differential equation that students need
to know how to solve are separable and
what does that mean well I can separate
the variables I can bluntly put all the
Y's on one side and all the x's on the
other so okay here's my differential
equation I'm going to get all my y's on
the left and all my x's on the right
hand side here we go in this step I'm
going to integrate both
sides on the left hand side I get a log
of the absolute value of y question we
often hear Tom and I often hear as well
do you need to really need to worry
about including that absolute value
symbol and the answer is yes especially
especially when we're dealing with
initial value problems yes you need to
include it we'll get rid of that or
we'll take care of eliminating those
absolute values as we eventually solve
this for why but yes you definitely need
it at this
step on the right hand side this is an
integration almost by inspection but
it's really substitution with u equal to
minus 2x I'll make maybe one more step
here so du is minus
2dx so that's an integration by
substitution I think that gives me a
min-2 eus 2X and this is the step right
here where this constant C has to come
in so where I find an anti-derivative of
the left side I find one of the right
side I've got to add that constant plus
C not a silly question but sometimes we
get this one well really when I find the
antiderivative on the left and one on
the right there's really two constants
agreed but I've combined all of those
and brought it over to the right hand
side
so now what do I do to isolate y what do
I do to get an explicit expression for y
well I'm going to exponentiate both
sides on the left hand side e and log
are inverse functions so I'm left with
the absolute value of y on the right
hand side kind of hard to see but that
expression minus
one2 eus 2x + C is all in the
exponent I'm going to use some
properties here of exponents so that
I've got an e to the c times this
expression and e to the C I will just
abbreviate or use some other notation
and call that a different constant
C1 now how do I isolate
why well I think about this
as that argument can be either positive
or
negative and if it's positive in my mind
I'm On One branch of this solution or
one piece of the solution if it's
negative I'm on another piece of the
solution so it's either going to be a
plus or minus y when I evaluate that
absolute value and we will incorporate
that sign into this constant C1 and
write that as a different constant
C2 now I didn't ask for this but I
actually created or included in the
solution files here the slope field
associated with this differential
equation and teachers students you might
think about this does this make sense
with the differential equation up above
um can you justify can you reason out
that this equation for y can you draw
that on this slope field it might not be
fine enough for some of some of this but
but does it make sense that that
equation somehow fits in this slope F so
we'll leave that as kind of an open
question here's B let's try this one
this is kind of a cool one too I like
this one so do ydx is e to xus 2 over
the cosine of Y I've got to separate the
variables I'm going to bring all the Y's
to the left once again all the xes to
the right I'm going to integrate both
sides here we go the integral of the
cosine of Y that's an integration by
inspection one of our common
anti-differentiation rules that's just
the sign of Y on the right hand side
integrate term by term so that's just e
to x minus 2X and here's where my
constant C comes
in and this isn't so bad I mean I want
to solve for why in order to do that I'd
have to take the inverse sign of both
sides so here's some things to think
about and then I'm going to hand it over
to Tom here I was thinking about this I
drew the slope field for this and I was
thinking about this and well wait a
minute what's the domain of this
function do I need to worry about that
or or do I need to think about that
here's a question for the chat
Allison what's got to be
true about this argument right there
what has to be true about that argument
in order me for me to be able to
evaluate the inverse sign or the arc
sign and the reason that I thought a
little bit about this was because I
didn't think originally that this slope
field actually made
sense so I started out by thinking about
the
domain and I'm G to turn it over to
TA see if he can justify a little bit of
this with
technology okay all all right let's give
it a
shot okay I'll U go ahead and use
the uh 84 on this one I think okay
sounds good
and so I'm going to go to my y equals
menu and uh clear out y1 and remember on
this slope field program on the 84
that's y1 is where we'll store our
differential equation
uh and uh let's see this was actually a
fraction so I'm going to do this uh
Alpha of the XT Theta key get a nice
little fraction there and let's see I
think it was e to the X
yep minus two correct that's my
numerator and then down at the do excuse
me in the denominator we've got the
cosine of
Y again it's a very weird looking
expression to have in y1 but that's just
our placeholder for the differential
equation
expression and let me go ahead and clear
Y
2 and now I'm going to
um go back to the calculator screen and
bring up the slope field program now
since we had run the slope field program
before I can either pull it out of
programs but since there before if I
just hit enter it's going to run it
again with this new differential
equation in
there and hopefully we'll get a slope
field that looks not dissimilar to the
one
that okay it might look a little bit
different just because of the way the
lattice is laid out and stuff but this
isett good pretty good yeah and now uh
we're GNA check out and see if Steve's
solution makes sense going to go back to
the Y equals and go to Y
2 uh we didn't have an initial condition
uh so to get a particular solution I'm
going to need a pick a value uh for C uh
so what we could do let's see I think
the the general function was the inverse
sign of e to the
x -
2x
mhm and then inside the parentheses
that's where that plus C showed up so we
could put in a value here uh we've got
to be a little bit careful about the
plus C because as Steve said the domain
of the inverse sign that's going to be
limited to between Nega one and one
right whatever I have in here it's got
to be something that I'll get some
values between negative one and one um
so I'm going to put it let's see e to
the zero is already
one uh but we are am subtracting X so
you know what I'm going to put in maybe
one half cross my
fingers so that's one value of the C
that we could try out and let's graph
it and I don't think I picked a very
good value no I don't see it Tom I don't
see it it might be off the screen so
I'll tell you what let's go back to y
equals I I'm uh I'm G to make the the
plus C the C value just zero I think see
yeah let's try
that okay that's a perfectly good value
of C now whether that's going to be one
that we'll see some urve for oh you know
what we lost our slope field because I
did a new graph there so let me just run
the slope field again okay regenerate
it doesn't take too
long and now go to y
equals uh turn on Y2
again and now we'll graph and this time
it should graph on top of the slope
field oh there we go how about that yeah
great and it looks like a pretty good
fit though it's you know it's in a
pretty there's not that many line
segments there so something we could do
do is we could change our window to
maybe just being the first
quadrant so I'm going to make the X-Men
zero the Y men
zero now when I do that what I'm going
to need to do is run the slope field
again to generate a new slope field for
the first
quadrant go so when that runs Tom does
that turn off off all the other y's and
Y it does it automatically turns off all
the Y's and then it's just referring to
why not y1 itself isn't even selected it
just it uses y1 inside the program just
to evaluate what the slope should be
these little line segments okay and when
I go back to Y notice that Y2 was turned
off yep need to turn it back on and
graph again and I get a little bit more
of now I so you can see I change the
window a lot each time I change the
window I do need to run the slope field
program again but this is looking like a
pretty good
fit pretty
cool all
right
so did you want to look at anything else
on this Steve or should I just Let's see
we had a suggestion okay try a c of
minus5 okay be happy I don't know if
that'll work try that one and see if
that does you know what I'm going to
make got my
Y3 okay so let's make
uh I'll just go ahead and enter it again
since that plus C was inside the
parentheses so minus
2X and the suggestion was minus .5 yes
okay and let's graph that
hey nice and notice it has a bit wider
domain to it yes uhuh very cool great
and so you can see that you can actually
do multiple solutions for different
values of the constant on the same
backdrop of the slope field yep very
nice okay stop the share there back to
you Steve all right
deal here we go so there was one
question that came in here about that
constant C in that one problem we did
Tom I guess it was a we a change from a
c to a C1 to a C2 and the question is do
students have to do that on the exam and
and the answer is probably not uh
they're very we are very forgiving about
that c uh as it changes quote unquote
you can just keep the same symbol C you
don't necessarily have to have a
subscript I don't think students would
uh lose a point for that in my opinion
yeah and it was kind of it is kind of a
subtle thing I mean your C1 Steve I
don't know if you want to back up to
that page
um I mean your your
C1 is really an
arbitrary positive or non- negative
constant yeah so you got an absolute
value of y is is an arbitrary positive
constant times this e to the one2 well
that means y itself is either plus or
minus that arbitrary positive constant
well you could just say that's just an
arbitrary constant period which that's
your C2 so the C1 was arbitrary positive
but C2 is just plain arbitrary it took
care of both positive and negative and
that's why you were able to drop that
absolute value sign y I think the way on
the exam it kind of gets around that is
you you often have some initial
condition you have to work with and
that's the thing that resolves the
matter and then it's cut and dryed
whether you're using the positive or
negative branch of
your correct all right okay I hope we
can I hope we do have the opportunity to
look at one of those here let's see um
let me take a look at C we'll solve one
more or maybe two more quick
differential equations and then we'll
take a look at an initial value problem
if we can
once again I'm going to get all the Y's
on one side of the equation all the x's
on the right hand side I'm going to
integrate both sides this one's pretty
easy this is a one one of our
anti-differentiation rules common
anti-differentiation rules that's an arc
tangent on the right hand side this is
again as a substitution with = x^2 + 1
and du = 2x
DX so let's see I integrate on the left
hand side add the tangent inverse on the
right hand side I had this expression in
X there's my plus C and again not too
bad to solve for why uh what's the
inverse of the inverse tangent well the
tangent I could take the tangent of both
sides and here's my expression can I
take the tangent of any
argument yeah I guess I can so I don't
have to worry too much about that domain
um does that function y fit nicely into
this slope field well again we'll kind
of lead that uh for teachers and
students to think about I provided that
slope field here you might try to draw
this function by looking at that slow
field good so see that tangent function
it will have just some isolated places
where it's not defined right
because but there won't be some whole
stretch of numbers where it's undefined
I think that's what yep all right well
let's take a look at three here we're
going to solve some initial value
problems um I like this one this
produced kind of a weird slope feeli
although this did not ask to do that I
had to see what was going on here so
here's the differential equation dydx is
x + 1 / the of XY we have this condition
y of 1 is equal to four I'm going to
start out in the usual way here's all my
y's on that side here's the x's on the
right hand side and looking ahead just a
little bit um how am I going to be able
to integrate that well the technique is
to write that as as two separate terms
or two separate fractions and to write
each of those terms as X raised to some
power so I did that over here now I'm
going to integrate both
sides don't make a mistake on this one
Tom on the left hand side 2/3 wider the
three halves on the right hand side we
integrate term by term and here's the
plus c um it is at this step now right
at this next step is where we recommend
that students use the initial condition
look can you go on and solve this for
y and then once you have isolated why
use the initial condition at that point
absolutely but in my opinion not only is
that harder but I think it introduces
more chance of an error so the best
thing to do is once you have found the
anti-derivative of both sides included
Your plus C use the initial condition
right away so here's the yal 4 here's
the xal 1 and saving a little bit of
time here I think that gives me a c
equal to
8/3 so I went back up here plugged in a
c equal to 8/3 and I multiply both sides
of that by three Hales is an expression
almost with Y
isolated how do I do that well how do I
get y all alone well I raise both sides
of this equation to the 2/3 power and
here's kind of a weird looking
expression or
why now here's what I did down at the
bottom and I may need a little help with
this
Tom so when I drew this slope field I
was kind of I I didn't know what was
going on initially
here but I only got a slope field in
quadrant one and
three I wasn't quite sure why so we'll
leave that as a question question for
participants why isn't there any slope
field for this differential equation in
quadrants 2 and
four
h well the second thing I did was I
wanted to actually draw this particular
solution curve that goes through the
0.14 so here it is here's the
0.4 and it looks like
this so my next question
is where X is equal to
zero is that actually on the solution
curve or should there be an open circle
there I
wonder so that's something else you can
think
about Allison we'll let you monitor the
chat and let us know if we get any
answers to those questions okay or those
are good ones maybe for :l tomorrow
morning in
class I liked that one Tom take a look
at could
be oh this has some good things in it
too so here we have dydx is equal to y -
2 1 + x^2 we have another initial value
problem y of 1 is three this is a
separable differential equation I can
get all the Y's on one side all the x's
on the other and this one involves the
log of an absolute value and let me make
sure that I do this correctly so on the
left hand side this is the log of the
absolute value of Yus 2 do you need the
absolute value yes you do you have to
worry about that on the right hand side
this is a common anti-differentiation
formula this is the arc tangent tangent
inverse of X and here's my plus
C okay at this step at this point we
want to use the initial condition what
was it y of 1 is equal to 3 so Y is
equal to 3 there's the xal to 1 and that
gives me a c = to minus piun 4 but it's
also told me something
else when this is the way that I think
about this when I plug in that y equal
to 3 this argument here this argument
inside the absolute value is
positive and that tells me I am on the
branch of this
solution where that argument is
positive if you think about for example
the graph of the log of the absolute
value of x just think about that I'm
going to scribble a little
bit if you think about that
graph it's not defined that function is
not defined at x equals z and there are
two pieces two branches to
that one branch corresponds to where X
is positive one corresponds to where X
is
negative in my mind that's kind of
what's going on
and I need the branch in this case where
the argument is
positive that means in my next step I
can simply drop the absolute
values how do you take the absolute
value of a positive argument it is just
that argument I don't need that extra
negative
sign so okay I've got the log of Yus 2
is equal to this expression with an arc
tangent over there I'm going to solve
this for y by first exponentiating and
then adding a two to both sides there's
my expression for y and there's some
cool things going on here I
think in this slope field and this
equation uh there's the SL f it kind of
looks like that the that
the line segments have slope zero at X
two wonder if I could justify that I
think I could if I look back up at the
differential
equation and I drew in this curve this
curve that satisfies the initial
condition of
13 now don't be misled here if you look
at this very
quickly it seems like there might be
some horizontal asmp tootes on the graph
of
y and it's seems like their two
horizontal ASM tootes might be at yal 2
and yal
4 but they're
not and so a question is can you find
the horizontal Asm
tootes on this graph of
y I wonder if the slope field might be
just a little bit misleading in finding
those okay we'll leave that as an open
question I like that one have we got
time for one more I think
so maybe a couple
more all right and C holy Toledo This is
complicated on the right hand side I'm
going to bring all the Y's to the left
all the X's to the right hand side oh
it's not so bad I guess the integral of
e to the x is just e to the X and this
isn't bad on the left hand side either
that's a simple substitution with Y = 1
+ y^2 I believe the derivative of of U
is 2 ydy the 2y is part of the integr
product except for that constant two so
we should be able to do this I integrate
both sides there's my constant C I use
the initial condition it turns out that
c is equal to
one so I'm going to go back up here and
I'm going to plug in a c equal one
there's my expression in X and Y can I
solve that can I isolate y and by the
way sometimes you cannot
not here I think I can I'm going to
square both sides I'm going to subtract
one from both sides and then I'm going
to take the square root of both sides
whoops there's my expression for y and I
did the same sort of thing down here I
drew a slope field and I drew in the
solution curve that goes through that
point what was it what was that initial
condition again I think it was Zero
theare of
three so there's another question I have
here you know as I looked at that
solution curve it looks an awful lot
like y equal e to the X yes granted it
does not go through the 01 but it still
looks an awful lot like y equal e to the
X and I wonder if you can make an
argument to explain that to justify that
why does that solution curve look a lot
like yal e to thex boy lots of open
questions here you're G to have to be
writing lots to Curtis this week I think
all right I'm going to take a look at
four Tom and maybe we'll call it a night
this is a good one here so you Ju Just
wanted to throw out that uh you know
that problem you had uh a couple back
where when you did the slope field it
only showed up in the first quadrant yes
thir quadrant yes I I believe in the
chat uh Wanda Burns pointed out that
that square root of the product of X and
Y
might be the culprit there and indeed it
is yep very good one yep there's a
square root of x times Y and in the
second and the fourth quadrant that
product would be negative and of course
the square root would be undefined that
is exactly why we don't have any part of
slope field in there very cool very
cool all right this was a nice one and I
I actually discovered something Tom on
Mathematica when I was doing this one
and produced an extra graph here just
for the heck of it I'm the the first
thing I asked is to use technology to
sketch a slope field for the
differential equation so I don't know
maybe we could end if you want to take a
look at this one eventually but this is
what I got using
Mathematica and I use this slope field
to sketch the solution curve that goes
through the0 21 so there's the point on
the graph 21 and I did my very best to
sort of sketch following those line
segments following those line segments
to draw a rough sketch of the solution
curve something sort of very interesting
seems to be happening here I hope that I
can explain this before I end this
problem but it looks like as we go over
here in the third quadrant it looks like
that curve is
following the line yal
X that seems awfully strange to me but
it seems to be doing
that I was looking something up in
mathematic to try to make my slopes
feels give them a couple of more options
make them look a little nicer
and I happen to cross a an option that
allows you to draw well what I would say
are several solution Curves in here and
I don't know if you can see this Tom but
you might be able to see the green sort
of arrows in there those correspond to
my solution curve that I drew up above
that's the curve that goes through the
point 2 one that was kind of cool gives
me even a better picture of what's going
on and those arrows actually
those vectors actually represent those
little line seg very cool yeah I believe
somebody in the chat was asking about uh
using slope fields to do vectors and I
think those are called Direction fields
and similar idea there yeah very cool
yeah I think I think when I did that
when I produced that if I let me go back
to that for one
second I think when I produced that one
of the options that I gave
uh Mathematica was something that would
produce those vectors so that they all
have the same
length but there is another option that
allows you to adjust the the length of
the magnitude of that
Vector U so that can be useful in in
other
applications all right so we're still
working on the same problem let's see if
we can solve this differential equation
I'm going to bring all the Y's to the
left all the X's to the right hand side
well this doesn't look too bad I've
separated the variables simple
substitution on both sides U is equal to
minus X U is equal to minus y I get this
expression we bring in the plus C we Ed
the initial
condition and I found this value of C
now that was a little cumbersome for me
to
write so what I did is I just said well
that's my value of c and I'm going to
assume that and solve for y here so that
I didn't have to write it out all the
time I did that I exponentiated both
sides I took the log of both
sides and I think I am left
with this
expression I multiplied through by a
minus one by the way in order to make
that a little bit easier to work with
and there's my expression for y and what
I did here is I actually used that
exess explicit value of c and I thought
a little bit about the domain here and I
think I think the domain here might be
all reals
okay so I'm G to add this
graph to my slope field in son of a gun
it looks an awful
lot like what I drew on the previous
page and I wonder if we can justify this
part over here what's going on over
there can we justify that at all I
wonder so as X
increases
negatively what's happening inside here
well as X gets bigger and bigger let's
see this is going to be a small number
so that's
inconsequential as X gets bigger and
bigger
let's see if that's inconsequential then
this is just like the log of e to the
minus X and those are inverse function
right so that's just going to give me a
minus X so it's like I've got a minus
Min -
x which is just X and remember X is
negative and so this is really just the
line yal X over here in the third
quadrant that's really cool so I saw
that on the slope field I wasn't sure
that it really made sense but I think we
can actually justify
that pretty cool
Tom well Steve I was looking at your
original differential equation so up the
page there a bit yes
um on the line y equal
x
my derivative will be exactly one how
about that because my e to the negative
x e to the Y will cancel out and give me
one but Y equals X is a line of slope
one so it it's all kind of falls into
place so kind of cool how about that I
think we're just about out of time
Steve we we're going to post these
problems uh with Allison and Curtis
tomorrow we do have one overtime problem
I've seen several of these in the
Facebook group recently Tom and this is
a very common sort of free response
question so uh I've got this one here in
the OT portion of our presentation and
we'll post this question and answer
tomorrow great so great thank you very
much for joining us final wrap up before
we close up I just wanted to share the
last of the chat was um noting that the
vector Direction fields are very
informative for uh golf and in fact
there was some conversation about seeing
uh that type of it's fun to see uh part
of math teacher Job showing up on
TV how about that thanks Allison great
job thanks Tom thank you Steve okay very
good
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