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

To check whether the given function is a solution of the given equation, we first need to calculate the derivative of the function.

`y'=-sin x` 

Now we substitute that into the equation.

`-x sin x-2cos x ne x^3e^x`  

Obviously, the two sides of the equation are not equal, hence the given function is not a solution of the differential equation.