Linear ODE Forms

matrix M

nodal source

(x',y') = M(x,y)

M =

1	1
1 2	2

Eigenvalues:

λ

1

λ

2

=

1

2

(3+

3

)

1

2

(3-

3

)

Eigenvectors:

(

3

-1)x(t)+y(t)

(-1-

3

)x(t)+y(t)

Solution:

x(t)

1

2

(

c

1

(

2t



+1)+

c

2

(

2t



-1))

y(t)

1

2

(

c

1

(

2t



-1)+

c

2

(

2t



+1))

The fixed point

(0,0)

has many different stability types. A system rarely has an infinite number of solutions along a line in the

x(t)

,

y(t)

plane.

The fixed point

(0,0)

has many different stability types. Only two solutions for fixed points are possible in a linear first-order system: one with an equilibrium point at

(0,0)

, or infinitely many solutions along a line in the

x(t)

,

y(t)

plane.

Red dots represent fixed points or equilibrium points.

Dashed blue lines indicate straight-line solutions.

Dashed black lines indicate nullclines.