- #1
jey1234
- 38
- 0
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 [itex]P[/itex] such that [itex]P^{-1} AP[/itex] is a diagonal matrix where [itex]A[/itex] is the coefficient matrix of the system above.
[itex]P=[/itex]
\begin{pmatrix}
0&-1&-1\\
1&1&0\\
2&0&1
\end{pmatrix}
[itex]P^{-1} AP=[/itex]
\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?
[itex]U'=(P^{-1} AP)U[/itex]
\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}
[itex]U=[/itex]
\begin{pmatrix}
c_1e^{-3x} \\
c_2e^{3x} \\
c_3e^{6x}
\end{pmatrix}
[itex]X=PU[/itex]
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.