Although hazardous may be you can write the DE in term of Laplace Transform as...

, (1)

In that case the can be [virtually] computed using the 'Inversion Complex Formula' as...

(2)

... where c is a constant that lies on the right of all singularities of the complex function...

(3)

In any case the question has to be further examined because I can have made some erroneous procedures...

Kind regards