# Thread: Using eigenvectors and eigenvalues to solve DE in matrix form

1. ## Using eigenvectors and eigenvalues to solve DE in matrix form

Hi, my friend asked for help with a question and I'm wondering if there's a better method than mine by the question wording.

We have the matrix
$\displaystyle A =\left( \begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array}\right)$
and have to find the eigen values and vectors which are
3 with $\displaystyle \left( \begin{array}{c} 1 \\ 1 \end{array} \right)$
and
1 with $\displaystyle \left( \begin{array}{c} -1 \\ 1 \end{array}\right)$

The next part is to solve the set of equations

$\displaystyle u_1'' = 2 u_1' + u_2'$
$\displaystyle u_2'' = u_1' + 2 u_2'$
$\displaystyle u_1(0) = 0$
$\displaystyle u_1'(0) = 1$
$\displaystyle u_2(0) = 1$
$\displaystyle u_2'(0) = 0$
using the eigenvalues and eigenvectors.

As this is of the matrix form I solve this by writing characteristic equations with the eigenvalues as the exponentials
$\displaystyle u_1 = A e^{t} + B e^{3t} + C$
$\displaystyle u_2 = D e^{t} + E e^{3t} + F$

and substituting into the above equations to get six equations in A,B,C,D,E and F thus leading to
$\displaystyle u_1 = \frac{1}{6}\left( 3e^{t} + e^{3t} -4 \right)$
$\displaystyle u_2 = \frac{1}{6}\left( -3e^{t} + e^{3t} +8 \right)$
which is right!

However, I haven't used the eigenvectors only the eigenvalues, is there another (better ? ) method that I should be using?

Thanks!

2. Hi

You can write the set of equations as
$\displaystyle U'' = AU'$ where $\displaystyle U = \left( \begin{array}{c} u_1 \\ u_2 \end{array} \right)$

A is diagonalisable into the diagonal matrix $\displaystyle D =\left( \begin{array}{cc} 3 & 0 \\ 0 & 1 \end{array}\right)$ through a transition matrix P such that $\displaystyle P^{-1}AP = D$

Let $\displaystyle V = \left( \begin{array}{c} v_1 \\ v_2 \end{array} \right) = P^{-1}U \Rightarrow U = PV$

$\displaystyle U'' = AU'$ is transposed into $\displaystyle PV'' = APV' \Rightarrow V'' = P^{-1}APV' = DV'$

Since D is diagonal $\displaystyle V'' = DV'$ is easily solved.

Then $\displaystyle U = PV$ is used to find $\displaystyle u_1$ and $\displaystyle u_2$
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