Proceed as in this example and find an explicit solution of the given initial-value problem. dy dx = y2 sin(x2), y(−4) = 1 6

is separable as

Integrating both sides gives

The integral on the right unfortunately has no elementary result (i.e. not in terms of Fresnel integrals). But we are given an initial value, so we can use the fundamental theorem of calculus to find a solution in terms of an integral.

Recall that

where .

Let be the antiderivative of . The theorem allows us to write

So the solution to the ODE is (upon integrating both sides)

Just to confirm this is valid, differentiating both sides gives

so all is well.

We can go on to solve for directly by taking the reciprocal of both sides, ending up with

the correct answer is x/3+10≥32

hope this !