Because it does not have constant coefficients, I expect you need to look for a series solution...
How to solve x^2 y'' - (x^2 +2x)y' + (x+2)y = x^3 e^x
What do we do when there is no hint given?
how to guess the probable solution, can we use the method used in the method of undetermined coefficients?
But this question appears in the book before the power series and other questions have a hint given. then assuming the hint as y1, y2 = vy1. But this question no hint is given, in this case how can we guess a solution.
Since the term on right side is on the form P(x)*exp(x) with P(x) a polynomial (in the present case P(x)=x^3), then we guess that y(x) could be on the form :
y(x) = p(x)*exp(x) where p(x) is a polynomial.
Considering the term on the left of the ODE, it appears that the degree of P(x) is the degree of p(x)+one unit.
So, we guess that y(x)=p(x)*exp(x) where p(x) is a polynomial of degree 2.
Then, the coefficients of this polynomial are obtained by identification.
On the other hand, it is not difficult to find that y=x and y=x*exp(x) are independant solutions of the ODE without the term on the right side.