# a very interesting differential equation

TLDRIn this lecture, the instructor explores a nonlinear differential equation, \( y y' = (y')^2 \). They emphasize its nonlinearity due to the quadratic term in the dependent variable and its derivatives. By substituting \( z = y' \), they transform it into a second-order differential equation. Various cases are solved using separation of variables, integrating factors, and substitutions, ultimately deriving solutions for different scenarios. The lecture provides a comprehensive analysis of solving complex nonlinear differential equations, highlighting key techniques and methods.

###### Takeaways

- π The script discusses a nonlinear second-order differential equation \( y'' = y' \cdot y' \) and its properties.
- π It identifies the equation as autonomous and nonlinear due to the quadratic term \( y' \cdot y' \).
- π A substitution method is introduced by setting \( Z = y' \) to simplify the equation to a new form involving \( Z \) and its derivatives.
- π§© The script uses the chain rule to transform the equation into a form that resembles the derivative of a function squared, \( \frac{d}{dx}(Z^2) \).
- π By recognizing the derivative of \( Z^2 \), the equation is further simplified to \( 2Z' = Z^2 + C \), where \( C \) is a constant.
- π The solution process involves considering different cases for the constant \( C \), including \( C = 0 \), \( C > 0 \), and \( C < 0 \).
- π For \( C = 0 \), the method of separation of variables is used to find the solution involving natural logarithms.
- π When \( C > 0 \), the solution involves the arctangent function and the natural logarithm of secant.
- π For \( C < 0 \), partial fraction decomposition is used, leading to an exponential form of the solution.
- π The final solutions for different cases are expressed in terms of exponential and logarithmic functions, showcasing the diversity of solutions for differential equations.
- π The script concludes with a comprehensive solution that includes integrals and substitutions to solve for \( y \) in terms of \( x \).

###### Q & A

### What type of differential equation is discussed in the transcript?

-The differential equation discussed is a nonlinear differential equation, specifically an autonomous differential equation where the dependent variable and its derivatives make the equation nonlinear.

### Why is the given differential equation considered nonlinear?

-The equation is considered nonlinear because it includes a term that involves the product of derivatives (y' * y'), making it quadratic in the dependent variable Y and its derivatives.

### What substitution is introduced to simplify the original differential equation?

-The substitution introduced is setting Z equal to Y', which simplifies the original equation by reducing it to a second-order nonlinear differential equation.

### How is the chain rule applied in the context of this differential equation?

-The chain rule is applied by recognizing that the right-hand side of the equation, Z * Z', can be expressed as the derivative of Z^2 with respect to X, simplifying the equation further.

### What is the solution approach when the constant 'a' equals zero?

-When 'a' equals zero, the solution approach involves separating variables, integrating both sides, and solving for Y, leading to a solution involving the natural logarithm of the variable X.

### How is the arc tangent function used in the solution for the case when 'a' is positive?

-For the case when 'a' is positive, the arc tangent function is used after separating variables and integrating, leading to a solution for Z in terms of the tangent of a linear function of X.

### What is the significance of renaming constants during the solution process?

-Renaming constants, such as replacing B with e^B, helps simplify the expression and makes it easier to manipulate and solve the differential equation.

### How does the method of separation of variables help in solving the equation?

-Separation of variables allows for the differential equation to be split into simpler parts that can be integrated individually, leading to a more straightforward solution for Y.

### What final form does the solution take in the case when 'a' is positive with a minus sign?

-The final solution in this case involves the natural logarithm of a quotient involving exponential functions, specifically expressing Y in terms of a natural logarithm that includes a difference of squares in the denominator.

### What is the overall strategy for solving the original third-order nonlinear differential equation?

-The overall strategy involves making a substitution to reduce the order of the equation, applying the chain rule, using separation of variables, and carefully managing constants to integrate and solve for Y, depending on the value of the constant 'a'.

###### Outlines

##### π Exploring a Nonlinear Differential Equation

This paragraph introduces the differential equation \( y \cdot y' = (y')^2 \), highlighting its nonlinearity due to the quadratic nature of the term involving \( y' \). The discussion covers the equation's classification as autonomous and nonlinear, and an initial approach to solving it by substitution. The author then introduces a new variable \( z = y' \), transforming the equation into a second-order nonlinear equation. A key insight involves recognizing a pattern resembling the chain rule, allowing further simplification.

##### π Separation of Variables and Antiderivatives

The author proceeds with the solution by applying the method of separation of variables. The transformed equation \( 2 \cdot \frac{dz}{dx} = z^2 \) is separated into terms involving \( z \) and \( x \). After integrating both sides, the equation is simplified, yielding an expression for \( z \), which corresponds to \( y' \). The paragraph concludes by finding the antiderivative of this expression to derive a function for \( y \), completing the solution for the case when the constant \( a \) is zero.

##### π Handling the Case with Positive \( a \)

In this paragraph, the author tackles the more complex scenario where the constant \( a \) is positive. Using separation of variables again, the differential equation is transformed and integrated. The antiderivative for the left-hand side involves the arctangent function, and after some algebraic manipulation, the equation for \( z \) (which equals \( y' \)) is obtained. Finally, the author derives the solution for \( y \) by integrating the obtained expression for \( y' \), resulting in a solution involving the natural logarithm of the secant function.

##### βοΈ Decomposing the Fraction for the Minus Case

This paragraph focuses on the scenario where \( a \) is positive, but the differential equation involves a minus sign. The author begins by separating the variables and then addresses the integral on the left side through partial fraction decomposition. After deriving the antiderivative, the equation is further simplified by exponentiating both sides. The solution is expressed in terms of \( Z \) and eventually transformed back to \( y' \). The paragraph concludes by presenting the integrated solution for \( y \), which includes a logarithmic expression.

##### π§© Final Simplification and Conclusion

In this concluding paragraph, the author combines the solutions from the previous cases. By further simplifying the expression for \( y \), involving natural logarithms and exponentials, the final form of the solution is derived. The author manipulates the equation to achieve a more elegant expression, suitable for the original differential equation. This final step completes the discussion and provides a neat, closed-form solution to the complex nonlinear differential equation introduced at the beginning.

###### Mindmap

###### Keywords

##### π‘Differential Equation

##### π‘Nonlinear

##### π‘Autonomous

##### π‘Substitution

##### π‘Chain Rule

##### π‘Separation of Variables

##### π‘Anti-Derivative

##### π‘ArcTan

##### π‘Partial Fraction Decomposition

##### π‘Natural Logarithm

##### π‘Tangent

##### π‘Exponential Function

###### Highlights

Introduction of a nonlinear differential equation: y' = y' * y'.

Explanation of why the equation is nonlinear due to the quadratic term y' * y'.

Substitution introduced: setting Z = y' to simplify the equation.

Transformation of the original differential equation into a second-order differential equation.

Identification of the right-hand side as a product of a function and its derivative, implying the use of the chain rule.

Introduction of a constant 'a' to account for the possible addition of a constant in the solution.

Solution of the differential equation in the case where a = 0, using separation of variables.

Derivation of the solution for y in the case when a = 0, resulting in y being expressed as a logarithmic function.

Introduction of the case where a > 0, with the equation involving an arctan function.

Derivation of the solution for y when a > 0 and a plus sign in the equation, leading to a solution involving a natural logarithm of the secant function.

Introduction of the final case where a > 0 and a minus sign in the equation.

Application of partial fraction decomposition to solve the differential equation in the final case.

Derivation of the solution for y in the final case, involving the natural logarithm of a ratio of exponential functions.

Final simplification of the solution for y in the last case, resulting in a compact logarithmic expression.

Conclusion of the lecture with a well-structured and simplified final solution for y.

###### Transcripts

## Browse More Related Video

Reducible Second Order Differential Equations, Missing Y (Differential Equations 26)

Separation of Variables // Differential Equations

Linear versus Nonlinear Differential Equations

Solving A Non-Linear Differential Equation

AP Calculus AB Crash Course Day 5 - Integration and Differential Equations

Separable First Order Differential Equations - Basic Introduction

5.0 / 5 (0 votes)

Thanks for rating: