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Arran's question at Yahoo! Answers (linear system)

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
Here is the question:

I'm trying to solve the coupled differential equations: dx/dt=x+4y and dy/dt=-5x+5y, but if you put the coefficients of x and y into a matrix, it gives complex Eigenvalues. So how do I solve the equations? Thanks for your help.
Here is a link to the question:

How do you solve a pair of coupled differential equations where the matrix of coeffs has complex Eigenvalues? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
Hello Arran,

The system can be written as $$\begin{bmatrix}{x'}\\{y'}\end{bmatrix}=A \begin{bmatrix}{x}\\{y}\end{bmatrix}\mbox{ with } A=\begin{bmatrix}{\;\;1}&{4}\\{-5}&{5}\end{bmatrix}$$
The eigenvalues of $A$ are $3\pm 4i$ (simple) so, $A$ is diagonalizable on $\mathbb{C}$. Now find an eigenvector $v_1$ associated to $3+4i$ and another one $v_2$ associated to $3-4i$. If $P=[v_1\;\;v_2]$ then, $$P^{-1}AP=D=\begin{bmatrix}{3+4i}&{0}\\{0}&{3-4i}\end{bmatrix}$$ The exponential matrix of $A$ is $e^{tA}=Pe^{tD}P^{-1}$. The general solution of the system is $$\begin{bmatrix}{x}\\{y}\end{bmatrix}=e^{tA} \begin{bmatrix}{C_1}\\{C_2}\end{bmatrix}\quad (C_1,C_2\in\mathbb{R})$$

P.S. Here you can test your result for $e^{tA}$

MatrixExp ({{t,4t},{-5t,5t}}) - Wolfram|Alpha
 
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