I need to solve the following PDE. $$u_{xx}-u_{tt} + au+be^{ct}=0, \ \ \ \ a,b,c\in R \ \ \ \ \ \ (1)$$
I took this approach: I am looking for a change of variables to transform the equation to $$u_{xx}-du_{tt} + f_1(x,t)=0. \ \ \ \ \ \ \ \ \ \ (2)$$ In fact I want to omit the term "$au$" from the equation and then solve the new equation. It would be appreciated if someone could help me.
I understood that there is no way to transfer Eq. (1) to Eq. (2). In order to solve Eq. (1), first we solve the homogeneous Eq. with respect to (1) i.e.
$$u_{xx}-u_{tt} + au=0$$
which is linear Kelin-Gordon equation associated with quantum mechanics. The solution of the eq. can be found in some handbooks. The solution for the in-homogeneous part of Eq. (1) is in the form $u=Ae^{ct}$ and $A$ can be calculated by substituting it in Eq. (1).