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

i think the correct option is the second option.