For a nonzero constant prove that

is a particular solution of the differential equation

Find the general solution of the equation

*Proof.* The general solution of is . So, we have particular solutions given by and (taking and , respectively). The Wronskian of and is

So, we have the functions and of Theorem 8.9 given by

This means we have a particular solution of the non-homogeneous equation given by

Using this theorem we can compute the general solution of

We have

I got -1/6x * cos(3x) +1/18 sin(3x) for the integral solution. However the constant of sin(3x) is anyways absorbed in c2, so I’m not sure why the solution in the book has it explicitly set to -1/18.

Same here