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

To determine whether the given function is a solution of the given differential equation, we can find the derivative of the function and check if it satisfies the equation.
To find the derivative of y = x^2e^x , use the product rule:
(fg)' = f'g + fg'
Here, f = x^2 and f' = 2x , and g = e^x and g' = e^x .
So y' = 2xe^x + x^2e^x = x(2 + x)e^x .
The left-hand side of the given equation will then be
xy' - 2y =2x^2e^x + x^3e^x- 2x^2e^x = x^3e^x . This is exactly the same as the right-hand side of the given equation, which means y(x) = x^2e^x is a solution.
The function y(x) = x^2e^x is a solution of the given differential equation.