Once You know a particular solution of the 'complete' DE, in order to solve the problem You have to find the general solution of the 'incomplete' DE that is...

(1)

This linear DE is 'Euler type' and its solution has the form . It is easy to verify that the only value of that satisfies (1) is so that the general solution of (1) is...

(2)

... and the general solution of the 'complete' DE is...

(3)

Kind regards