Question

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

Answer 1

This equation is separable, but $\sin(x^2)$ doesn't have an elementary antiderivative. However, we can use the fundamental theorem of calculus to find a solution in terms of an integral:

$\dfrac{\mathrm dy}{\mathrm dx}=\sin(x^2)$

$\implies y(x)=y(\sqrt\pi)+\displaystyle\int_{\sqrt\pi}^x\sin(u^2)\,\mathrm du$

$\implies y(x)=4+\displaystyle\int_{\sqrt\pi}^x\sin(u^2)\,\mathrm du$