Solve the given initial-value problem.
y''' − 2y'' + y' = 2 − 24ex + 40e5x, y(0) =
1
2
, y'(0) =
5
2
, y''(0) = −
11
2
the values for the first 3 derivatives derivatives of y are 1/2, 5/2, -11/2 respectively

For the corresponding homogeneous ODE,

the characteristic equation is

which admits the characteristic solution,

Assume a particular solution of the form

( because a constant solution is already accounted for by ; because both and are accounted for)

Substituting the derivatives of into the ODE gives

So the particular solution is

With the given initial conditions, we find

and so

ok then what happened.