Simplifying a solution that has complex eigenvalues

[Jamin2112]

Homework Statement

I'll give an example.

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

Homework Equations

Assume a solution of the form x=$e^rt for these type of problems.

Euler's formula: e^bi = cosb + isinb

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.

 

[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