- #1

- 986

- 9

## Homework Statement

Express the general solutoins of the system of equations in terms of real-valued functions.

**x**'= [1 0 0; 2 1 -2; 3 2 1]

**x**(I wrote the matrix MATLAB-style)

## Homework Equations

The coolest equation ever: e

^{ib}=cosb +

*i*sinb

## The Attempt at a Solution

Assume

**x**=

**e**

__R__^{rt}(the underlined r is an eigenvector)

Determinate[1-r 0 0; 2 1-r -2; 3 2 1-r]=0 --> r = 1, 1+2

*i*, 1-2

*i*

r = 1 --> [0 0 0; 2 1 -2; 3 2 0](

__r___{1}

__r___{2}**)**

__r___{3}^{T}=(0 0 0)

^{T}

--->

__= (2 -3 2)__

**R**^{1}^{T}

I do the same with the other eigenvalues, and come up with 3 eigenvectors:

__= (2 -3 2)__

**R**^{1}^{T},

__= (0 1 -__

**R**^{2}*i*)

^{T},

__= (0 1__

**R**^{3}*i*)

^{T}.

By Superpsition, the full solution will be

**x**(t)=C

_{1}e

^{t}(2 -3 2)

^{T}+ C

_{2}e

^{t}

^{e2it}(0 1 -i)

^{T}+ C

_{3}e

^{t}e

^{-2it}(0 1 i)

^{T}

= e

^{t}[ C

_{1}(2 -3 2)

^{T}+ C

_{2}(cos(2t)+

*i*sin(2t))(0 1 -

*i*)

^{T}+ C

_{3}(cos(-2t)+

*i*sin(-2t))(0 1

*i*)

^{T}]

................................. This somehow simplifies to the answer in the back of the book, C

_{1}e

^{t}(2 -3 2)

^{T}+ C

_{2}e

^{t}(0 cos2t sin2t)

^{T}+ C

_{3}e

^{t}(0 sin2t -cos2t)

^{T}. I don't understand the simplification process. Yes, I know the imaginary numbers just get absorbed into the constants; but I can't figure out the rest.