Question

Finding General Solution with Matrices

Answer 1

Find the general solution to:

\[X' = \left[\begin{matrix}1 & -1 \\ 0 &-1\end{matrix}\right]X +\left(\begin{matrix}t \\ e^{3t}\end{matrix}\right)\]

Answer 2

i would be able to do one of these problems if the \[\left(\begin{matrix}t \\ e^{3t}\end{matrix}\right)\] part wasn't there... I don't know what to do with the extra part that is added on

Answer 3

do I have to use Variation of Parameters?

Answer 4

so i got my complementary solution to be \[C_1e^{-t}\left(\begin{matrix}1 \\ 2\end{matrix}\right)+ C_2e^{t}\left(\begin{matrix}1 \\ 0\end{matrix}\right)\] that's what i have so far

to find the particular solution, do i need to use variation of parameters somehow? this is where i'm lost

Answer 5

and when i say "complementary solution", i mean the general solution to the homogeneous equation

Answer 6

omg! what's that ? Linear Algebra combined with DE?! the two most difficult math subjects has joined as one! I'll quit my math class if my teacher starts to talk about that in class, seriously

Answer 7

I understand the term PI and CF very well, but I have no idea with that sucky thing really

Answer 8

@caozeyuan LA+DE is actually a class. http://www.google.com/search?q=Linear+Algebra+of+differential+equations

And that also pulls up lots of notes on the topic of them together.

Answer 9

this is actually for a DE class, but i know that this is related to LA a lot too.

Also to solve this, i took the inverse of the 2x2 matrix and multiplied it by 1/(det) and then took the integral. this was the method i found online for nonhomogeneous linear systems, but i remember using learning to use Variation of Parameters in class