Undetermined coefficients to find general solution to system

In summary: You are left with cwte^t = e^t, so c = 1. Then you have to figure out the values of a and b to make the remaining terms equal.In summary, the conversation is about solving a system of differential equations using the Method of undetermined coefficients. The person is having trouble finding the particular solution and asks for help. Another person suggests a trial solution but it is incorrect. They discuss how to find the correct values and equate them to solve for c, a, and b.
  • #1
mslodyczka
4
0
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
 
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  • #2
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 ]
 
  • #3
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
 
  • #4
mslodyczka said:
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

mslodyczka said:
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.
 

1. What is the undetermined coefficients method?

The undetermined coefficients method is a technique used to find the general solution to a system of linear differential equations. It involves assuming a particular form for the solution and then solving for the coefficients that satisfy the differential equations.

2. When is the undetermined coefficients method applicable?

The method is applicable when the system of differential equations is linear and has constant coefficients. It can also be used for systems with variable coefficients if certain conditions are met.

3. How do you determine the particular form for the solution?

The particular form for the solution is determined by examining the form of the non-homogeneous term in the differential equations. The form should be chosen in such a way that it does not overlap with the form of the homogeneous solution.

4. Can the undetermined coefficients method be used for non-linear systems?

No, the undetermined coefficients method is only applicable for linear systems of differential equations. For non-linear systems, other techniques such as the variation of parameters method may be used.

5. What is the general process for using the undetermined coefficients method?

The general process for using the undetermined coefficients method is as follows:
1. Identify the form of the non-homogeneous term in the differential equations.
2. Choose a particular form for the solution that does not overlap with the homogeneous solution.
3. Substitute the particular form into the differential equations and solve for the coefficients.
4. Combine the particular solution with the homogeneous solution to get the general solution.

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