# Solve 3rd order ode using variation of parameters

## Homework Statement

Solve using variation of parameters
y''' - 2y'' - y' + 2y = exp(4t)

## Homework Equations

Solve using variation of parameters

## The Attempt at a Solution

I got the homogenous solutions to be 1, -1, and 2.

So, y = Aexp(t) + Bexp(-t) + Cexp(2t) + g(t)

I got W{exp(t), exp(-t), exp(2t)} = 6exp(2t)

My professor did quite an unsatisfactory job explaining variation of parameter for high order equations (spent an hour and 30 minutes trying to do 1 problem, and still didn't finish it correctly). So I just need to be pointed in the right direction.

Thanks

## The Attempt at a Solution

CompuChip
Homework Helper
Actually, I got -6 exp(2t), but maybe I made a sign error in my calculation.

So the general idea is now to replace the ith column in the matrix by (0, 0, exp(4t)) and calculate the determinant |Wi| of that, for i = 1, 2, 3.
Then if you calculate
$$c_i(t) = \int \frac{|W_i|}{|W|} \, dt$$
the general solution is
$$y(t) = c_1(t) e^{t} + c_2(t) e^{-t} + c_3(t) e^{-4t}$$

LCKurtz
Homework Helper
Gold Member

## Homework Statement

Solve using variation of parameters
y''' - 2y'' - y' + 2y = exp(4t)

## Homework Equations

Solve using variation of parameters

As a learning exercise, fine. But I assume you know nobody in their right mind would use variation of parameters to find a particular solution, eh?