Phase Plane - Example of A Linear System

Example of A Linear System

A two-dimensional system of linear differential equations can be written in the form:

\begin{align} \frac{dx}{dt} & = Ax + By \\ \frac{dy}{dt} & = Cx + Dy \end{align}

which can be organized into a matrix equation:

\begin{align} & \frac{d}{dt} \begin{pmatrix} x \\ y \\ \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \\ \end{pmatrix}\begin{pmatrix} x \\ y \\ \end{pmatrix} \\ & \frac{d\mathbf{x}}{dt} = \mathbf{A}\mathbf{x}. \end{align}

where A is the 2 × 2 coefficient matrix above, and x = (x, y) is a coordinate vector of two independent variables.

Such systems may be solved analytically, for this case by integrating:

although the solutions are implicit functions in x and y, and are difficult to interpret.

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