Writing the DE equation as...

(1)

... the general solution of the homogeneus equation [i.e. the equation in which is ...] is...

(2)

In the case You have proposed is , so that is solution of the homogeneous equation and that can produce some problems. In almost all such cases the solution of the problem requires the use of Laplace Transform. Writing (1) in term of LT we obtain...

(3)

... so that is...

(4)

The particular solution You are searching is the inverse LT of (4) and loocking at the 'manual' we find that this 'particular solution' is [excluding errors of me ] ...

(5)

Kind regards