`y = x^3` Determine whether the function is a solution of the differential equation `xy' - 2y = x^3e^x`

In order to check if the given function is solution of th given differential equation, we first need to find the derivative of the function.

`y'=3x^2`

Now we plug the into the equation.

`x cdot 3x^2-2cdotx^3=3x^3-2x^3=x^3 ne x^3e^x`  

Since the left and the right hand side of the equation are not equal, the given function is not a solution to the given differential equation.