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.

Read more about this topic:  Phase Plane

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