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2. Consider M(x, y) + N(x, y) = 0. (a) Suppose we can find a function F example of such an equation and solve it. (b) Suppose we can find functions and z so that (u o z)(x, y) is an integrating factor that transforms the given equation into an exact equation. What must be true about M and N? = F(x, y) so that Fx = M and Fy = N. Give an

2. Consider M(x, y) + N(x, y) = 0. (a) Suppose we can find a function F example of such an equation and solve it. (b) Suppose we can find functions and z so that (u o z)(x, y) is an integrating factor that transforms the given equation into an exact equation. What must be true about M and N? = F(x, y) so that Fx = M and Fy = N. Give an

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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2. Consider M(x, y) + N(x, y) · = 0. (a) Suppose we can find a function F example of such an equation and solve it. F(x, y) so that F = M and F, = N. Give an (b) Suppose we can find functions u and z so that (uo z)(x, y) is an integrating factor that transforms the given equation into an exact equation. What must be true about M and N? (c) Continuing from part (b), suppose we can find functions u and z so that (uoz)(x, y) is an 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 u in part (c), given u(zo) = Ho, 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. (g) What sources did you use for this exercise?
Transcribed Image Text:2. Consider M(x, y) + N(x, y) · = 0. (a) Suppose we can find a function F example of such an equation and solve it. F(x, y) so that F = M and F, = N. Give an (b) Suppose we can find functions u and z so that (uo z)(x, y) is an integrating factor that transforms the given equation into an exact equation. What must be true about M and N? (c) Continuing from part (b), suppose we can find functions u and z so that (uoz)(x, y) is an 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 u in part (c), given u(zo) = Ho, 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. (g) What sources did you use for this exercise?
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