Couldnt you let L(y) = y''' + p(t)y'' + q(t)y' + r(t)y = f(t)For the last solution,
You can let z=y'
The general solution to homogenous equation is:
Then you can find particular via Lagrange's Variation of Parameters techiqnue.
Then you need to integrate the entire solution for z tto get y.
and y_h = c_1y_! + c_2y_2 + c_3y_3 and y_p = v_1y_1+v_2y_2+v_3y_3
Then v_1', v_2, and v'_3 have to satisfy some auxillary condition, and you can use Crameris Rule to get the solution.