This an inhomegeneous linear ODE with constant cefficients.

So if f(x) is the general solution to the homogeneous equation:

y"(x) - 3y'(x) - 2y(x) = 0,

and g(x) is a particular soulution of

y"(x)=3y'(x) + 2y(x) + sinh (x)

then the general solution is y(x) = f(x) + g(x).

You can find a particular solution using a trail solution:

g(x) = A sinh(x) + B cosh(x).

Plug this into the equation and by equating coefficients of sinh and cosh

on both sides of the equation you will end up with a pair of linear equations

fo A and B. Which if I'm not mistaken has solution A=0, B=-1/3.

RonL