Undetermined coefficients to find general solution to system

Hi,
I'm having a bit of trouble with a problem here.

The question is: Use the Method of undetermined coefficients to Find the general solution to th system:

dx/dt = y + e^t
dy/dt = -2x + 3y + 1

I've got the homogenous solution fine, however I'm having a bit of difficulty with the particular solution.

I used xp = [ ctwe^t + ue^t ] where w was [1,1]^T but i know this trial doesnt include the 1 term and is therefore incorrect.

Can someone let me know what I'm supposed to do as a trial solution in this case. It's not explained in my notes, and I've looked online but to no avail.

Thanks!
Mike

HallsofIvy
Homework Helper
What you give will work for the et part- though you really don't need the et- tet is enough. For the "1" you need to add a constant : say
xp = [ (cte^t+ d)w ]

hi,
thanks for your help. I still cannot manage to get the solution. Is there anyway you can show some working I'd appreciate it?

What I did:

Let xp = [ (cte^t+ d)w ]

therefore xp (differentiated) = cwe^t + cwte^t + 0

Then Ax(t) + g(t)
is cwte^t + [ b + 1 , -2a + 3b ]^T + [ e^t , 1 ]^T

now we are supposed to equate to get the values of c, a and b but I cannot see how to do this...
Thanks

HallsofIvy
Homework Helper
Hi,
I'm having a bit of trouble with a problem here.

The question is: Use the Method of undetermined coefficients to Find the general solution to th system:

dx/dt = y + e^t
dy/dt = -2x + 3y + 1

I've got the homogenous solution fine, however I'm having a bit of difficulty with the particular solution.

I used xp = [ ctwe^t + ue^t ] where w was [1,1]^T but i know this trial doesnt include the 1 term and is therefore incorrect.

Can someone let me know what I'm supposed to do as a trial solution in this case. It's not explained in my notes, and I've looked online but to no avail.

Thanks!
Mike

hi,
thanks for your help. I still cannot manage to get the solution. Is there anyway you can show some working I'd appreciate it?

What I did:

Let xp = [ (cte^t+ d)w ]

therefore xp (differentiated) = cwe^t + cwte^t + 0

Then Ax(t) + g(t)
is cwte^t + [ b + 1 , -2a + 3b ]^T + [ e^t , 1 ]^T

now we are supposed to equate to get the values of c, a and b but I cannot see how to do this...
Thanks

What is A? You didn't mention that before. Is that the matrix mutliplying x in your original equation? If so you set that equal to the derivatives on the left. Because et is a solution to the homogenous equation, the terms involving tet will cancel out.