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:

`(\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`