Solveing differential equations system using diagonal matrix

[jey1234]

Homework Statement

Solve this system of differential equations

\begin{equation}
x'_1=5x_1 + 2 x_2 - x_3 \\
x'_2=-2x_1 + x_2 - 2x_3 \\
x'_3=-6x_1 - 6 x_2
\end{equation}

Homework Equations

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?

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

\begin{equation}
u'_1=-3u_1 \\
u'_2=3u_2 \\
u'_3=6u_3
\end{equation}

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

$U=$
\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.

 

[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. :)