Homogeneous Linear System of Coupled Differential Equations

matrix entries

a

0.5

b

-0.75

c

0.1

d

0.5

′

x

′

y

= 

0.5	-0.75
0.1	0.5



x

y



λ

1

= 0.50+0.27

λ

2

= 0.50-0.27

ν

1

= 

0.94+0.00

0.00-0.34



ν

2

= 

0.94+0.00

0.00+0.34



This Demonstration shows the solution paths, critical point, eigenvalues, and eigenvectors for the following system of homogeneous first-order coupled equations:

x'=ax+by

,

y'=cx+dy

.

The origin is the critical point of the system, where

x'=0

and

y'=0

. You can track the path of the solution passing through a point by dragging the locator. This is not a plot in time like a typical vector path; rather it follows the

x

and

y

solutions. A variety of behaviors is possible, including that the solutions converge to the origin, diverge from it, or spiral around it.