This is non-homogeneous ODE, so you should be looking at the characteristic equation of the homogeneous version:
which has solution , so the general solution of the homogeneous equation is:
and a particular integral of the inhomogeneous equation would be:
So the general solution is:
Which is all academic because the problem in the attachment is clearly not this one (but the real question could be sloved this way as well if you know what the particular integral will look like).