# Find the solution of the differential equation xy'=y+x^2\sinx that satisfies the given initial conditions below?

## $y \left(\setminus \pi\right) = 0$ I know it's in $\frac{\mathrm{dy}}{\mathrm{dx}} + P \left(x\right) y = Q \left(x\right)$ form, but what exactly is the integrating factor? Like what is "exp" in $I = \exp \left(\setminus \int P \left(x\right) \mathrm{dx}\right)$? (People use this a lot when mentioning the integrating factor, but I learned it as $I \left(x\right) = {e}^{\setminus \int P \left(x\right) \mathrm{dx}}$...)

Apr 25, 2018

Refer to the Explanation.

#### Explanation:

$\exp$ in $\exp \left(\int P \left(x\right) \mathrm{dx}\right)$ means exponential function.

So, $\exp \left(\int P \left(x\right) \mathrm{dx}\right)$ is just another way to denote

${e}^{\int P \left(x\right) \mathrm{dx}}$.

I hope I've cleared your doubt!

By the way, I presume that you know how to solve the given diff.

eqn., so, I don't solve it!