(c) Continuing from part (b), suppose we can find functions and z so that (uoz)(x, y) is an μ integrating factor that transforms the given equation into an exact equation. Construct a differential equation with respect to u to solve for u. (d) Using the ODE with respect to μ in part (c), given (zo) = μo, when does a unique solution exist? (e) Suppose z = x² + y³. Find u from part (c), if possible. (f) Construct an example of an equation that can be solved using the integrating factor in part (e) and then solve this equation.
(c) Continuing from part (b), suppose we can find functions and z so that (uoz)(x, y) is an μ integrating factor that transforms the given equation into an exact equation. Construct a differential equation with respect to u to solve for u. (d) Using the ODE with respect to μ in part (c), given (zo) = μo, when does a unique solution exist? (e) Suppose z = x² + y³. Find u from part (c), if possible. (f) Construct an example of an equation that can be solved using the integrating factor in part (e) and then solve this equation.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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b, c, d, e, f only
Transcribed Image Text:The image contains a set of mathematical problems related to differential equations and integrating factors. Below is the transcribed text for educational purposes:
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### Differential Equations Problems
#### Problem Set
**(c)** Continuing from part (b), suppose we can find functions \(\mu\) and \(z\) so that \((\mu \circ z)(x, y)\) is an integrating factor that transforms the given equation into an exact equation. Construct a differential equation with respect to \(\mu\) to solve for \(\mu\).
**(d)** Using the ODE with respect to \(\mu\) in part (c), given \(\mu(z_0) = \mu_0\), when does a unique solution exist?
**(e)** Suppose \(z = x^2 + y^3\). Find \(\mu\) from part (c), if possible.
**(f)** Construct an example of an equation that can be solved using the integrating factor in part (e) and then solve this equation.
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These problems are designed to guide students through the process of finding and applying an integrating factor to transform a differential equation into an exact equation, solving for the integrating factor, and applying it to an example for solution verification. Students are expected to show their work and understanding of differential equations and mathematical transformations.
![**Title: Solving Differential Equations: Exact Equations and Integrating Factors**
**Consider**
\[ M(x, y) + N(x, y) \cdot \frac{dy}{dx} = 0. \]
**Problem Statement:**
**(a)** Suppose we can find a function \( F = F(x, y) \) so that \( F_x = M \) and \( F_y = N \). Give an example of such an equation and solve it.
**(b)** Suppose we can find functions \( \mu \) and \( z \) so that \( (\mu \circ z)(x, y) \) is an integrating factor that transforms the given equation into an exact equation. What must be true about \( M \) and \( N \?
### Explanation:
1. **Exact Equations:**
An equation of the form \( M(x,y)dx + N(x,y)dy = 0 \) is exact if there exists a function \( F(x, y) \) whose partial derivatives satisfy:
\[ \frac{\partial F}{\partial x} = M \]
\[ \frac{\partial F}{\partial y} = N \]
To solve such an equation, one must:
- Find \( F \) by integrating \( M \) with respect to \( x \) and \( N \) with respect to \( y \).
- Ensure that the equations \( F_x = M \) and \( F_y = N \) are consistent.
2. **Integrating Factor:**
For a non-exact equation, an integrating factor \( \mu(x, y) \) can be found. The integrating factor is a function that, when multiplied with the original differential equation, makes it exact. The condition for this transformation involves functions of \( M \) and \( N \) and their partial derivatives.
### Example for Part (a):
Consider the differential equation:
\[ (2xy + y^2)dx + (x^2 + 2xy + 1)dy = 0. \]
To determine if it's exact, we check whether there exists a potential function \( F(x, y) \):
- \( F_x = 2xy + y^2 \)
- \( F_y = x^2 + 2xy + 1 \)
By integrating \( M = 2xy + y^2 \) with respect to \( x \),](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64590550-d676-4b57-9a24-39401c3be4ef%2F018dc9b0-ba0d-4a51-b364-63f321c03fd8%2F5zvzocf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving Differential Equations: Exact Equations and Integrating Factors**
**Consider**
\[ M(x, y) + N(x, y) \cdot \frac{dy}{dx} = 0. \]
**Problem Statement:**
**(a)** Suppose we can find a function \( F = F(x, y) \) so that \( F_x = M \) and \( F_y = N \). Give an example of such an equation and solve it.
**(b)** Suppose we can find functions \( \mu \) and \( z \) so that \( (\mu \circ z)(x, y) \) is an integrating factor that transforms the given equation into an exact equation. What must be true about \( M \) and \( N \?
### Explanation:
1. **Exact Equations:**
An equation of the form \( M(x,y)dx + N(x,y)dy = 0 \) is exact if there exists a function \( F(x, y) \) whose partial derivatives satisfy:
\[ \frac{\partial F}{\partial x} = M \]
\[ \frac{\partial F}{\partial y} = N \]
To solve such an equation, one must:
- Find \( F \) by integrating \( M \) with respect to \( x \) and \( N \) with respect to \( y \).
- Ensure that the equations \( F_x = M \) and \( F_y = N \) are consistent.
2. **Integrating Factor:**
For a non-exact equation, an integrating factor \( \mu(x, y) \) can be found. The integrating factor is a function that, when multiplied with the original differential equation, makes it exact. The condition for this transformation involves functions of \( M \) and \( N \) and their partial derivatives.
### Example for Part (a):
Consider the differential equation:
\[ (2xy + y^2)dx + (x^2 + 2xy + 1)dy = 0. \]
To determine if it's exact, we check whether there exists a potential function \( F(x, y) \):
- \( F_x = 2xy + y^2 \)
- \( F_y = x^2 + 2xy + 1 \)
By integrating \( M = 2xy + y^2 \) with respect to \( x \),
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