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...