Continuous dynamical systems are described by systems of ordinary differential equations:

** x'** =

where `t` is the independent variable (usually considered as the time for evolution), and

` x` =

The system is linear if ` f`(

x'_{1} = |
a_{11}(t)x_{1} |
+ a_{12}(t)x_{2} |
+ ... + a_{1n}(t)x_{n} |

x'_{2} = |
a_{21}(t)x_{1} |
+ a_{22}(t)x_{2} |
+ ... + a_{2n}(t)x_{n} |

: | |||

x' =_{n} |
a_{n1}(t)x_{1} |
+ a_{n2}(t)x_{2} |
+ ... + a(_{nn}t)x_{n} |

and may be abbreviated as ` x'` =

The general solution of a system ` x'` =

` x`(

where **v**_{1}(`t`), **v**_{2}(`t`), ..., ` v_{n}`(

Since ` x` →

In the special case the coefficient functions are constants: `a _{ij}`(

(`e ^{λt}v`)'

= (

=

=

=

Thus we conclude the following.

Suppose **v**_{1}, **v**_{2}, ..., ` v_{n}` is a basis of eigenvectors of

` x`(

Example From the eigenvalues 5, 15 and the eigenvectors (1, -2), (2, 1) found in this example for the matrix

[ | 13 | -4 | ], |

-4 | 7 |

we conclude that the system

x' = |
13x |
- 4y |

y' = |
- 4x |
+ 7y |

has two (linearly independent) solutions

[ | x_{1}(t) |
] = e^{5t}[ |
1 | ], [ | x_{2}(t) |
] = e^{15t}[ |
2 | ], |

y_{1}(t) |
-2 | y_{2}(t) |
1 |

and the general solution for the system is

[ | x(t) |
] = c_{1}e^{5t}[ |
1 | ] + c_{2}e^{15t}[ |
2 | ] = [ | c_{1}e^{5t} + 2c_{2}e^{15t} |
]. |

y(t) |
-2 | 1 | - 2c_{1}e^{5t} + c_{2}e^{15t} |

Example We try to find the solution of the system

x' = |
x |
+ 5y |

y' = |
- 2x |
+ 3y |

satisfying the initial condition

`x`(0) = -1, `y`(0) = 14.

Based on this example, we find the general solution

`x`(`t`) = (1 - 3`i`)`c`_{1}`e`^{(2 + 3i)t} + (1 + 3`i`)`c`_{2}`e`^{(2 - 3i)t},

`y`(`t`) = 2`c`_{1}`e`^{(2 + 3i)t} + 2`c`_{2}`e`^{(2 - 3i)t}

of the system. By (see this note)

`e`^{(2 + 3i)t} = `e`^{2t}(cos3`t` + `i`sin3`t`)

and the similar formula for `e`^{(2 - 3i)t}, we may rewrite the general solution as

`x`(`t`) = `ae`^{2t}(cos3`t` + 3sin3`t`) + `be`^{2t}(- 3cos3`t` + sin3`t`),

`y`(`t`) = `ae`^{2t}(2cos3`t`) + `be`^{2t}(2sin3`t`),

where `a` = `c`_{1} + `c`_{2}, `b` = `i`(`c`_{1} -`c`_{2}).

To find the solution satisfying the initial condition, we substitute the initial condition to the general solution and get

`a` - 3`b` = -1,

2`a` + 2`b` = 12.

Then we get `a` = 5 and `b` = 2 and the special solution

`x`(`t`) = `e`^{2t}(- cos3`t` + 17sin3`t`),

`y`(`t`) = `e`^{2t}(10cos3`t` + 4sin3`t`),

satisfying the initial condition.

In the example above, we saw that if we take `a`_{1} and `a`_{2} to be real numbers, then we get the real-valued general solution in case of complex eigenvalues. In general, suppose a *real* matrix ` A` has a conjugate pair of

`c`_{1}`e`^{(μ + iν)t}(` u` +

=

as part of the general solution of the system ` x'` =

[previous topic] [part 1] [part 2] [part 3] [next topic]