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2. Consider M(x, y) + N(z,y) = 0. (a) Suppose we can find a function F = F(z.9) so that F, = M and F, = N. Give an example of such an equation and solve it. (b) Suppose we can find functions u and : so that (uo 2)(1, y) is an integrating factor that transforms the given equation into an exact equation. What must be true about M and

2. Consider M(x, y) + N(z,y) = 0. (a) Suppose we can find a function F = F(z.9) so that F, = M and F, = N. Give an example of such an equation and solve it. (b) Suppose we can find functions u and : so that (uo 2)(1, y) is an integrating factor that transforms the given equation into an exact equation. What must be true about M and

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 12CR
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2. Consider M(r, y) + N(1, y) # = 0. (a) Suppose we can find a function F = F(r,y) so that F, = M and F, = N. Give an example of such an equation and solve it. (b) Suppose we can find functions p and z so that (u o 2)(z. y) is an integrating factor that transforms the given equation into an exact equation. What must be true about M and N? (c) Continning from part (b), suppose we can find functions p and z so that (uo2)(1. 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 p. (d) Using the ODE with respect to u in part (e), given p(20) = uo, when does a unique solution exist? (e) Suppose z= r* + y. 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(r, y) + N(1, y) # = 0. (a) Suppose we can find a function F = F(r,y) so that F, = M and F, = N. Give an example of such an equation and solve it. (b) Suppose we can find functions p and z so that (u o 2)(z. y) is an integrating factor that transforms the given equation into an exact equation. What must be true about M and N? (c) Continning from part (b), suppose we can find functions p and z so that (uo2)(1. 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 p. (d) Using the ODE with respect to u in part (e), given p(20) = uo, when does a unique solution exist? (e) Suppose z= r* + y. 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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