Maths question (desperately need help) (1 Viewer)

[kitty123]

Hi,
I need to find the particular solution to the following system of differentical equations using the method of undetermined coefficients.

dx/dt = y + e^t

dy/dt = -2x + 3y + 1

The problem is the forcing terms of the two diff equations are different, that is one is polynomial and the other an exponential. I have no idea what you "guess" the particular solution set to be in this situation.

Any help will be GREATLY appreciated!

Thank you!

 

[Templar]

Write y' for dy/dt and x' for dx/dt, differentiate the second equation we get
y''=-2 x'+3 y'
which gives us
y''-3y'+2y=-2e^t

LHS is a simple ODE that you should be able to solve, and the RHS is relatively easy to guess (try Ate^t or something like that).

 

[kitty123]

thank you Templar.. i can see what you mean

But i really need to find the particular solution the system using "the method of undetermined coeffients".

 

[Templar]

You are using the method of undertermined coefficients to solve the inhomogeneous RHS. You're simply putting it into a form where you can use that method. You need to reconcile x and y, otherwise you can't really solve the equations. It's like trying to solve simultaneous equations by dealing with each equation individually, you need to use both for it to work.

 

[kitty123]

Yeah, I guess I can do it that way and argue I DID use the method of undetermined coefficients.. even though I am not sure if we are allowed to change the first order system to higher orders.

For example if you had a system like
dx/dt = x + 3y + 2e^(4t)
dy/dt = 2x + 2y - e^(4t)
the particular solution would be of the form c*t*w*e^(4t) + u*e^(4t) where w is an eigenvector of the coefficient matrix of the system and 4 is an eigenvalue. The "coefficients" we have to find are c and the elements of vector u.. say u = (a,b)^T.. so a, b and c