`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.