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 (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? (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) solution exist? = lo, when does a unique (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.

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