Ordinary differential equations Definition

Homogeneous linear system with constant coefficients

Builds on System of linear differential equations

Definition

Let $n \ge 1$ and let $A$ be a real $n \times n$ matrix. The homogeneous linear system with constant coefficients given by $A$ is the system of linear differential equations $x'(t) = A\, x(t)$ for an unknown differentiable function $x : \mathbb{R} \to \mathbb{R}^n$ .

Remarks

Its solutions are $x(t) = e^{t A} x(0)$ . See the matrix exponential as the solution. If $A v = \lambda v$ , then $x(t) = e^{\lambda t} v$ is a solution. The eigenvalue method builds all solutions from such solutions. Example: for $A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ , the solutions are rotations: $x(t) = \left(a \cos t + b \sin t, -a \sin t + b \cos t\right)$ .

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Order of a differential equation definition
Linear differential equation of order n definition
System of linear differential equations definition
Homogeneous linear system with constant coefficients definition

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Tag	`0VJF99CZ0Z`
URI	`m2://0VJF99CZ0Z`
Kind	definition
Writing	W0 stub
MSC	34A30
Defines	Homogeneous linear system with constant coefficients
Uses	System of linear differential equations

Homogeneous linear system with constant coefficients

Definition

Remarks

Reading path

Where this leads

Further reading

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