# Simplifying a solution that has complex eigenvalues

1. Apr 27, 2010

### Jamin2112

1. The problem statement, all variables and given/known data

I'll give an example.

Ex: x'=[-1/2 1; -1 -1/2]x.

2. Relevant equations

Assume a solution of the form x=\$ert for these type of problems.

Euler's formula: ebi = cosb + isinb

3. The attempt at a solution

|A-rI|=0

---> r= -1/2 +/- i

---> x= e-t/2 ( C1(cost + isint)(1 i)T + C2(cos(-t) +isin(-t))(1 -i)T )

I understand that I can simplify a little with the fact that sin(-t)=sin(t) and cos(-t)=-cos(t), but I don't understand how to simplify it all the way to

C1e-t/2 (cost -sint)T + C2e-t/2(sint cost)T,

which is the answer in the book.

So, explain.

2. Apr 27, 2010

### lanedance

so basically your 2 linearly independent solutions are

$$\textbf{x}_1 = e^{-t/2}(cos(t) + i.sin(t))(\begin{matrix} 1 \\ i \end{matrix})$$
$$\textbf{x}_2 = e^{-t/2} (cos(-t) + i.sin(-t))(\begin{matrix} 1 \\ -i \end{matrix})$$

note that any linear combination of these will also be a solution, so perhaps you could try taking 2 linear combinations that simplify things... making sure they are still linearly independent