Second order linear system and power series: Differential Equations

clarineterr

1. The problem statement, all variables and given/known data
Find a third degree polynomial approximation for the general solution to the differential equation:

[tex]\frac{d^{2}y}{dt^{2}}[/tex] +3[tex]\frac{dy}{dt}[/tex]+2y= ln(t+1)

2. Relevant equations
Power series expansion for ln(t+1)


3. The attempt at a solution

The system to the corresponding homogeneous equation [tex]\frac{d^{2}y}{dt^{2}}[/tex] +3[tex]\frac{dy}{dt}[/tex]+2y = 0

is y(t) = k₁e^-t +k₂e^-2t

Then I guessed[tex]\frac{ at^{3}}{3}[/tex]-[tex]\frac{bt^{2}}{2}[/tex]+ct as a solution for the original equation. Plugging this in I got a=1/2, b=2,c=2/3

But then I still have the t[tex]^{4}[/tex], t[tex]^{5}[/tex] terms, etc left in the equation. Im not quite sure how a third degree polynomial can be a solution to this equation.

NSAC

Because it will be an approximation not really the solution itself.