Answer
$y=x^2e^x-5x^2$ IS a solution to the given differential equation.
Work Step by Step
We will find $y'$ and put it into the given differential equation along with $y$.
$$y' = (x^2e^x-5x^2)'=(x^2e^x)'-(5x^2)' = (x^2)'e^x+x^2(e^x)'-10x=2xe^x+x^2e^x-10x.$$
Now let us put this into the given equation:
The Left side:
$$xy'-2y=x(2xe^x+x^2e^x-10x)-2(x^2e^x-5x^2) = 2x^2e^x+x^3e^x-10x^2-2x^2e^x+10x^2=x^3e^x.$$
The Right side is just
$$x^3e^x.$$
We see that both sides are equal to eachother which means that $y=x^2e^x-5x^2$ IS a solution to the given differential equation.
