Consider M(x, y) + N(x, y) = 0. (a) Suppose we can find a function F = F(x, y) so that Fa = M and Fy = N. Give an example of such an equation and solve it.
Consider M(x, y) + N(x, y) = 0. (a) Suppose we can find a function F = F(x, y) so that Fa = M and Fy = N. Give an example of such an equation and solve it.
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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Transcribed Image Text:(c) Continuing from part (b), suppose we can find functions \( \mu \) and \( z \) so that \( (\mu z)(x,y) \) 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.
Transcribed Image Text:**Consider \( M(x, y) + N(x, y) \cdot \frac{dy}{dx} = 0 \).**
**(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 \( N \)?
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