Unfortunately you cannot use the method of undermined coefficients when the ODE does not have constant coefficients. You will need to use the variation of parameters.

Variation of parameters - Wikipedia, the free encyclopedia

since you are given the solutions to the homogeneous problem, we can solve for the particular solution.

1st. Put equation is standard form

2st. Assume solution of the form

3nd.Calculate the Wronskian

Now

and

Now

and

Now

Putting all of this together gives