For your second problem, just treat it as you would a 2nd order ODE
ie solve the homogeonous eqn y''''+8y''-9y=0 by trying y=me^x.
This gives you a quartic auxilliary eqn to solve which in turn gives you the complementary function.
then find the particular integral by trying the solution y=Exe^x+Fsinx+Gcosx to find E,F,G.
add the complementary function to the particular integral to get the general solution.
Can't help with your first problem, sorry...