# Homework Help: Solveing differential equations system using diagonal matrix

1. Dec 10, 2012

### jey1234

1. The problem statement, all variables and given/known data
Solve this system of differential equations

x'_1=5x_1 + 2 x_2 - x_3 \\
x'_2=-2x_1 + x_2 - 2x_3 \\
x'_3=-6x_1 - 6 x_2

2. Relevant equations
3. The attempt at a solution
This is my first time solving a problem like this and I just wanted to make sure if what I did was correct.

Using the eigenvalues and eigenvectors, I found the matrix $P$ such that $P^{-1} AP$ is a diagonal matrix where $A$ is the coefficient matrix of the system above.

$P=$
\begin{pmatrix}
0&-1&-1\\
1&1&0\\
2&0&1
\end{pmatrix}

$P^{-1} AP=$
\begin{pmatrix}
-3&0&0\\
0&3&0\\
0&0&6
\end{pmatrix}

----------

I am sure everything above is correct. So assuming it is correct, is the following process correct?

$U'=(P^{-1} AP)U$

u'_1=-3u_1 \\
u'_2=3u_2 \\
u'_3=6u_3

u_1=c_1e^{-3x} \\
u_2=c_2e^{3x} \\
u_3=c_3e^{6x}

$U=$
\begin{pmatrix}
c_1e^{-3x} \\
c_2e^{3x} \\
c_3e^{6x}
\end{pmatrix}

$X=PU$

I understand this is pretty long. I'd appreciate if someone can just give a quick look at the process and the substitutions I made (with U and U'). Thanks.

2. Dec 10, 2012

### jey1234

Never mind. I just realized I can verify by seeing if my solution satisfies the given system. And it does. So I guess that means my soultion is correct. :)