If y = A x e ∫ P d x is a solution of d y d x + P x y = Q x , then A ' ( x ) =

The solution is given as,

$y=A\left(x\right)e^{\int Pdx}$

The above equation is a linear differential equation.

The integral factor is,

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

The solution is,

$y\left(\mathrm{IF}\right)=\int Q\left(x\right)\times \mathrm{IF}dx$

$y{e}^{\int P\left(x\right)dx}=\int Q\left(x\right)e^{\int P\left(x\right)dx}dx$

Substitute the values in the above equation from given
expression.

$A\left(x\right)=\int Q\left(x\right)e^{\int P\left(x\right)dx}dx$

$\Rightarrow A^{\text{'}}\left(x\right)=Q\left(x\right)e^{\int P\left(x\right)dx}$