Dy/dx=sin(x^2) with the initial condition y(π)^1/2=4

This equation is separable, but doesn't have an elementary antiderivative. However, we can use the fundamental theorem of calculus to find a solution in terms of an integral:

look at the picture.

y-intercept for the function on a picture is 4 → point (0, 4).

y-intercept for the function

is for x = 0. substitute:

y-intercept is 8 → point (0, 8).