Undetermined coefficients to find general solution to system

[mslodyczka]

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 doesn't 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]

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

 

[mslodyczka]

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]

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 doesn't 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 e^t is a solution to the homogenous equation, the terms involving te^t will cancel out.