Well let me try (remember try :-))

You have the ODE

to solve. You have the complimentary solution

.

To find a particular solution you try

and see if you can find a and to satisfies your ODE with As you said

.

Next (no assumption here)

Substituting into your ODE you have

.

Now you know that some pieces together go to zero as both and satisfies the complementary ODE so

or

.

At this point you haveoneequation for the two unknows and - equation (1) undetermined. You could guess a and try and solve (1) but that could prove to be difficult. As you only have a single equation, i.e eqn. (1), you could split the equation into two pieces. For example,

.

.

and each can be reduced to a linear first order ODE, however, by setting

gives

.

What's nice about this split is you have twolinearequations for the two unknows and - there's no and ! So solve for and and then integrate.