Solving homogenous system with complex eigenvalues \(\displaystyle{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={2}{x}+{8}{y}\) \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}=-{x}-{2}{y}\) When

Kiara Haas

Kiara Haas

Answered question

2022-04-03

Solving homogenous system with complex eigenvalues
dxdt=2x+8y
dydt=x2y
When I solve the determinant of the matrix, I get λ=±2i. Then , I plug it in the matrix, and get the for the first eigenvalue, λ=2i:
0=(22i)x+8y
0=x(2+2i)y
This gives
x=822iy
x=(2+2i)y

Answer & Explanation

Ireland Vaughan

Ireland Vaughan

Beginner2022-04-04Added 14 answers

Note that 822i=8(2+2i)(22i)(2+2i)=8(2+2i)8=(2+2i),
so the two equations are saying the same thing (as they must, if the eigenvalues are correctly computed). So you can simply take, for example, y=1 and x=2+2i to get an eigenvector.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?