Thread: Solve DE using eigenvalues

Solve the following set of differential equations (using eigenvalues and eigenvectors) where u is a vector of dimension 2.

$\displaystyle \displaystyle \frac{du}{dt}=\left[ \begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array} \right]u$

My attempt: The eigenvalues of A are 1, 1. $\displaystyle \displaystyle \left[ \begin{array}{c} 0 \\ 1 \end{array} \right]$ is the only eigenvector. How do I proceed?

Let

$\displaystyle A=\begin{bmatrix}1 &0\\1 &1\end{bmatrix}.$

Your solution is $\displaystyle u=e^{tA}u_{0}.$

So you must compute $\displaystyle e^{tA}.$ But how? It turns out that with this particular $\displaystyle A,$ you can easily compute the $\displaystyle n$th power, thus enabling you to compute

$\displaystyle e^{tA}=\displaystyle\sum_{n=0}^{\infty}\frac{(tA)^ {n}}{n!}.$

Normally, you'd use the eigenvalue expansion. But, as you've noticed, you can't do that with this matrix. You could try to use generalized eigenvectors. Another approach is to try to compute the nth power directly, which I think you can do here. What do you get when you compute the nth power of A?