Can you solve (a-bx)y'+(c-dx)y-e=0 with a,b,c,d,e constants?

[JanisEB]

I'm having trouble solving the differential equation (a-bx)y'+(c-dx)y-e=0 with a,b,c,d,e constants.
I tried laplace transforming it, but then I end up with yet another differential equation in the laplace domain because of the xy and xy' terms.

 

[Simon Bridge]

So what methods have you ruled out?

(a-bx)y'+(c-dx)y-e=0
You really want to avoid using "d" as a constant label in differential equations.

The DE is 1st order, linear, and has form:
y' + p(x)y = q(x) if you put p = (c-dx)/(a-bx) and q=e/(a-bx)

More generally: f(x)y' + g(x)y = e

Have you looked for an integrating factor? - watch for funny integrals like the gamma function.

 

[JanisEB]

Thanks a lot! That works, but indeed I end up with a funny integral that does not have an analytical solution. I was hoping for an elegant solution, but I'll have to rely on my computer then.