The 'incomplete' DE is...
(1)
... and its 'characteristic equation' is...
(2)
... the solution of which are , so that the general integral of the (1) is...
(3)
... that is different from the solution of the starting post.
Now a particular solution of the 'complete' DE...
(4)
... You can verify is ...
(5)
... so that the general integral of (4) is...
(6)
Kind regards