Question

(Seemingly simple differentiation equation problem using linear algebra).

http://f.imgtmp.com/BMTZp.jpg

I would greatly appreciate if someone could show me how to solve this problem.

Thanks in advance!

Answer 1

i think it has to do with eugene values/vectors for the homogenos part

Answer 2

it is a homogeneous eqn, so that's all there is too it I think

Answer 3

so the eigenvalues are what?

Answer 4

just for the practice :) I think I come up with -5+- 5i

Answer 5

Evector = [ b/(L-a) , 1 ]

Answer 6

yup

Answer 7

First you need to find the eigenvalues:
\[\det (A -\lambda I) = -5\pm 5i\]

Answer 8

now put them in the system (A−λI)

Answer 9

and solve to find eigenvectors

Answer 10

What's A here?

Answer 11

A is the matrix

Answer 12

Wolfram alpha gets 0: http://www.wolframalpha.com/input/?i=det( {{-5,-5},{5,-5}}+-+(-5%2B5i){{1,0},{0,1}}) and I get -50 for the eigenvalue. If I'm right, how do I proceed now?

Answer 13

the Eigenvalues are complex, and we have them above.
so first step would be to get the correct eigenvalue (which amistre has already found)

Answer 14

Oh! That's why I'm getting 0! (Because using what he found makes the characteristic polynomial 0!) So, what's the step that follows finding the eigenvalue? The complex numbers confuse me.

Answer 15

oh boy, let me break out my notes for this one :P

Answer 16

well we gotta find the eigenvectors, so let's see if I remember how to do that

Answer 17

our e-values are\[\lambda_{1,2}=-5\pm5i\]so we need to solve the systems\[(A-\lambda_1 I)\vec\eta\]and\[(A-\lambda_2 I)\vec\eta\]

Answer 18

for \(\lambda_1\) we have the system... oh man sorry gotta go
sorry to do this to you but here is a really good link on this stuff
http://tutorial.math.lamar.edu/Classes/DE/ComplexEigenvalues.aspx