Exact Differential Equations
TLDRThis video tutorial explains the concept of exact differential equations in a detailed and engaging manner. It begins by defining an exact differential equation and its conditions, such as the continuity of partial derivatives in a connected region. The script then demonstrates the process of solving such an equation step by step, including taking partial derivatives, integrating with respect to x and y, and finding a function whose derivative matches the given equation. The final solution is obtained by setting the derived function equal to an arbitrary constant, simplifying the complex process into a clear and concise explanation.
Takeaways
- 📚 The video discusses solving an exact differential equation, a type of equation where a function of x and y and another function of x and y times y' are set equal to zero.
- 🔍 An exact differential equation is identified when the partial derivative of m with respect to y equals the partial derivative of n with respect to x.
- 📉 The solution involves finding a function whose derivative with respect to x is m and whose derivative with respect to y is n.
- 📈 The partial derivatives of the functions m and n must be continuous in the original variables on a connected region for the equation to be exact.
- 📝 The given example in the script has functions m and n that are already in a suitable form for differentiation.
- 🧩 By taking the partial derivative of m with respect to y, treating x as a constant, we get the derivative of y cosine x as cosine x and 2x e^y as 2x e^y.
- 📐 Similarly, taking the partial derivative of n with respect to x gives the derivative of sine as cosine, x squared as 2x, and the constant term disappears.
- 🔄 Since the partial derivatives of m and n with respect to y and x respectively are equal, the equation is confirmed to be exact.
- ⚙️ The next step is to integrate the function m with respect to x, which yields y sine x and x squared e^y, plus an arbitrary function of y.
- 🔑 The function's partial derivative with respect to y must equal the n part, which leads to determining the derivative of the arbitrary function h'(y) as -1.
- 🧮 Integrating h'(y) gives h(y) as -y, which is then plugged back into the function f(x, y) to complete the solution.
- 🎯 The final solution is implicitly defined and is set equal to an arbitrary constant C, which concludes the solution process.
Q & A
What is an exact differential equation?
-An exact differential equation is a type of differential equation where a function of x and y and another function of x and y, times y prime, are set equal to zero. The equation is exact if the partial derivative of one function with respect to y and the partial derivative of the other function with respect to x are equal.
How can you determine if an equation is exact?
-To determine if an equation is exact, take the partial derivative of the function M with respect to y and the partial derivative of the function N with respect to x. If these partial derivatives are equal, the equation is exact.
What are the conditions for the functions in an exact differential equation?
-The partial derivatives of the functions involved in an exact differential equation must be continuous on a connected region.
What is the first step in solving an exact differential equation?
-The first step is to identify the functions M(x,y) and N(x,y) and verify that the equation is exact by checking that the partial derivative of M with respect to y equals the partial derivative of N with respect to x.
How do you find the function F(x, y) in an exact differential equation?
-You find the function F(x, y) by integrating the function M(x,y) with respect to x, and then adding a function of y, which represents any function that vanishes when differentiated with respect to x.
What do you do after integrating M(x,y) with respect to x?
-After integrating M(x,y) with respect to x, you take the derivative of the resulting function with respect to y and set it equal to N(x,y). This helps to determine the function of y that was added during the integration.
How do you solve for the added function of y?
-To solve for the added function of y, take the partial derivative of the integrated function with respect to y and set it equal to N(x,y). This will allow you to solve for the function of y.
What is the final form of the solution to an exact differential equation?
-The final form of the solution to an exact differential equation is the function F(x, y) set equal to an arbitrary constant C. This implicitly defines the solution.
Why is the final solution to an exact differential equation implicitly defined?
-The final solution is implicitly defined because it is expressed as a function F(x, y) equal to an arbitrary constant C, rather than explicitly solving for y in terms of x or vice versa.
What is the general process for solving an exact differential equation?
-The general process involves verifying the equation is exact, integrating the function M(x,y) with respect to x, determining the function of y by comparing the partial derivatives, and then combining these to form the solution F(x, y) = C.
Outlines
📚 Introduction to Exact Differential Equations
The video begins with an introduction to exact differential equations, explaining that they involve two functions of x and y, and their derivatives with respect to x and y. The condition for an equation to be exact is that the partial derivative of the first function with respect to y must equal the partial derivative of the second function with respect to x. The video emphasizes the importance of these partial derivatives being continuous in the original variables over a connected region. The first part of the equation is presented, which includes functions m(x, y) and n(x, y), and the process of taking their partial derivatives to verify the equation's exactness is demonstrated.
🔍 Solving the Exact Differential Equation
The second paragraph delves into the process of solving the exact differential equation. It starts by integrating the function m with respect to x, which results in a new function f(x, y) that includes terms like y times sine x and x squared e to the power of y. The integration process is detailed, including the handling of the e to the y term, which requires an arbitrary function of y due to the integration with respect to x. The paragraph then discusses the integration of the derivative of f with respect to y, which leads to the determination of the function h'(y) and subsequently h(y). The integration of h'(y) reveals that h(y) is simply negative y. Finally, the video concludes by combining all the integrated parts to form the complete solution function f(x, y), which is then set equal to an arbitrary constant C to represent the general solution to the problem.
Mindmap
Keywords
💡Exact Differential Equation
💡Function of x and y
💡Partial Derivative
💡Connected Region
💡Integration
💡Derivative
💡Generic Function
💡Solution
💡Arbitrary Constant
💡Differentiate
Highlights
Introduction to exact differential equations and their definition.
Explanation of the condition for an equation to be considered exact: equality of partial derivatives of two functions.
Requirement for partial derivatives to be continuous in the original variables on a connected region.
Presentation of the given differential equation with functions m(x, y) and n(x, y).
Demonstration of taking partial derivatives of m with respect to y and n with respect to x.
Confirmation that the given differential equation is exact due to equal partial derivatives.
Process of finding a function whose derivative with respect to x is m and with respect to y is n.
Integration of m with respect to x to find the function f(x, y).
Introduction of a generic function of y to account for the integration with respect to x.
Differentiation of the integrated function with respect to y to match the n part of the equation.
Identification of h'(y) from the condition that the derivative of the function equals the n part.
Integration of h'(y) to find h(y), which is a function of y only.
Inclusion of h(y) into the function f(x, y) to complete the solution.
Setting the complete function equal to an arbitrary constant c for the final solution.
Summary of the process involving integration, differentiation, and setting to an arbitrary constant.
Invitation for viewers to post comments and ask questions for further clarification.
Transcripts
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