# Repeated eigenvalues

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1. May 7, 2016

1. The problem statement, all variables and given/known data
I want to solve this system

x' = $\left( \begin{array}\\ 7 & 1 \\ -4 & 3 \end{array} \right)$x + $\left( \begin{array}\\ t \\ 2t \end{array} \right)$

2. Relevant equations

3. The attempt at a solution

i found the eigenvalues to both be 5. The eigenvector is (1,-2) and the generalized eigenvector i found to be (0,1)

I'm confused on how to solve the non-homogeneous part, since I got repeated eigenvalues. Is the procedure the same? Can I use, say, variation of parameters to solve this?

Last edited: May 7, 2016
2. May 7, 2016

### stevendaryl

Staff Emeritus
I apologize for being dense, but I don't understand this notation. Is the expression on the right a 2x2 matrix? If so, it would make it more readable if you used Tex notation:

$\left( \begin{array}\\ 7 & 1 \\ -4 & 3 \end{array} \right)$

3. May 7, 2016

Yes, that's right. I'm not really familiar with Latex, I'll have to read up on it. I've edited my post, thanks

4. May 7, 2016

### Staff: Mentor

As written, this makes no sense. You can't add a 2 x 2 array to a 2 x1 array (column matrix).

This would make more sense if it were something like this:
$\vec{x'} = \begin{bmatrix} 7 & 1 \\ -4 & 3 \end{bmatrix}\vec{x} + \begin{bmatrix} 1 \\ 2 \end{bmatrix}$

Here $\vec{x}$ means $\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ and similar for its derivative.

5. May 7, 2016

Yes that's that I meant, sorry for the confusion. So could i set up a fundamental matrix with the homogeneous solutions and solve the non-homogeneous system with, say, variation of parameters? I tried and this is what I get:

x(t) = c1e5t$\left( \begin{array}\\ -1 \\ 2 \end{array} \right)$ + c2e5t$\left( \begin{array}\\ -t-1/2 \\ 2t \end{array} \right)$ - (t/25)$\left( \begin{array}\\ 1 \\ 18 \end{array} \right)$ + (1/125)$\left( \begin{array}\\ 3 \\ -26 \end{array} \right)$

I checked my solution on wolfram and it's slightly different, which annoys me.

6. May 7, 2016

### Staff: Mentor

Just check that your solution satisfies the system of diff. equations.

7. May 7, 2016

### Ray Vickson

Isn't there a factor $x$ missing on the right? Should you not be dealing with the system
$$\pmatrix{x_1'\\x_2'} = \pmatrix{7 & 1 \\ -4 & 3} \pmatrix{x_1\\x_2} + \pmatrix{t \\2t} ?$$
You can either plug in the matrix exponential in the solution
$${\mathbf{x}} = e^{At} \int_0^t e^{-A \tau} {\mathbf{f}}(\tau) \, d \tau$$
to your equation ${\mathbf{x}}'(t) = A {\mathbf{x}}(t) + {\mathbf{f}}(t)$, or else use the Laplace-transform method.

As for the matrix exponential: you have $A = P J P^{-1}$, where $J$ is the Jordan canonical form of $A$:
$$J = \pmatrix{5 & 1 \\0 & 5}$$
Furthermore, for any scalar $x$ we have $e^{Ax} = P e^{Jx} P^{-1}$, and $e^{Jx}$ is easy to determine; see
webpages on matrix exponentials.

8. May 7, 2016

I just checked, and its satisfied. Thanks!

9. May 7, 2016