## Calculus 10th Edition

$y=x^3$ is not a solution to given differential equation.
We will find $y'$ and put it into the given differential equation along with $y$. $$y'= (x^3)'=3x^2.$$ Putting this into the differential equation we get The Left side: $$xy'-2y= x\times 3x^2 -2x^3= 3x^3-2x^3=x^3.$$ The Right side: $$x^3e^x.$$ We see that The Left side is NOT equal to the Right side for all $x$ so $y=x^3$ is not a solution to this differential equation.