Solve the System of Differential Equations

Question

Solve the system:

dx/dt = -4y+e^-t

dy/dt=9x-e^-t

I wish I could provide more info but my problem is that I'm not sure what method I need to use for this. This is from a study guide for my class that covers a lot of sections and I do not remember which of the various methods we've learned that I need to use for this one.

Answer 1

John,

I will sketch this out for you. First you need to find the general solution of the homogeneous equations (i.e., without the e^(-t) terms on the right hand side). The best way to do this is to rewrite this as a matrix equation and find the eigenvalues/eigenvectors. In this case, the eigenvalues are a complex conjugate pair, so you'll want to re-express the eigenfunctions as sines and cosines. If you've done this correctly, you'll end up with an expression involving two undetermined constants.

Then you need to find a particular solution of the non-homogeneous equation, which is straightforward to do by the method of undetermined coefficients. Add this to the general solution to the homogeneous equation and you are home free!

Feel free to contact me individually if you'd like me to walk you through this in more detail (or other issues from your study guide!)

Raphael