Consider the following initial-value problem. (x + y)2 dx + (2xy + x2 − 2) dy = 0, y(1) = 1 Let ∂f ∂x = (x + y)2 = x2 + 2xy + y2. Integrate each term of this partial derivative with respect to x, letting h(y) be an

Suppose the ODE has a solution of the form , with total differential

This ODE is exact if the mixed partial derivatives are equal, i.e.

We have

so the ODE is indeed exact.

Integrating both sides of

with respect to gives

Differentiating both sides with respect to gives

so the general solution to the ODE is

Given that , we find

so that the solution to the IVP is

step-by-step explanation:

f(x)=4

step-by-step explanation: