WOLFRAM|DEMONSTRATIONS PROJECT

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.