i keep getting stuck in some problems of physics because the differential equation of the form

a dy/dx + by = c f(x)

keeps cropping up. is there any general method to solve this differential equation?

when f(x) is sinusoidal, we avoid it using j operators and get the solution but in my particular problem f(x) is exponential.

Yes, there is a general approach. More than one less-than-general approach. You may wish to Google around for "(first order) inhomogenous linear ODEs".

 

In this case, the important thing is that a sum of solutions to an ODE, is a solution to that ODE.

 

So for the most general answer you will want to give.

 

y(x) = u(x) + v(x)

 

Where u(x) is the a y(x) that satifies LHS=0 (the implicit sol'n).

And v(x) is a y(x) that satisfies y(x)=RHS and LHS=RHS (the particular sol'n).