Second order linear system and power series: Differential Equations

[clarineterr]

Homework Statement

Find a third degree polynomial approximation for the general solution to the differential equation:

$$\frac{d^{2}y}{dt^{2}}$$ +3$$\frac{dy}{dt}$$+2y= ln(t+1)

Homework Equations

Power series expansion for ln(t+1)

The Attempt at a Solution

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

is y(t) = k₁e^-t +k₂e^-2t

Then I guessed$$\frac{ at^{3}}{3}$$-$$\frac{bt^{2}}{2}$$+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$$^{4}$$, t$$^{5}$$ terms, etc left in the equation. I am not quite sure how a third degree polynomial can be a solution to this equation.

 

[NSAC]

I am not quite sure how a third degree polynomial can be a solution to this equation.

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