Consider the following initial$-$value problem.

$y^{\text{'}}+4xy=x^3{e}^{x^2},y(0)=5$

Find the coefficient function $P(x)$ when the given differential equation is written in the standard form $\frac{dy}{dx}+P(x)y=f(x).$

$P(x)=$

Find the integrating factor for the differential equation.

$e^{{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}P(x)dx}=$

Solve the given initial$-$value problem.

$y(x)=$

Give the largest interval I over which the solution is defined for the given initial condition. $($Enter your answer using interval notation.$)$

$I=$