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.