To determine whether the given function is a solution of the given differential equation, we first need to find the derivative of the function.
Now we plug that into the equation.
As we can see, after simplifying the left hand side we get the right hand side of the equation. This means that the given function is a solution of the given differential equation.
Of course, this is only one of the solutions. The general solution of this equation is `y=x^2(c+e^x).`
The image below shows graphs of several such functions for different values of `c.` The graph of the function from the beginning is the green one.