# Math Insight

### Applet: Solution of linear system versus time and on phase plane with eigenvectors

Control panel (Show)
Solution (Show)
Eigenvalues and eigenvectors (Show)

Illustration of the solution to a system of two linear ordinary differential equations. The system is of the form $\diff{\vc{x}}{t} = A\vc{x}$ with prescribed initial conditions $\vc{x}(0)=\vc{x}_0$, where $\vc{x}(t)=(x(t),y(t))$. The components of the solution $x(t)$ (thick blue curve) and $y(t)$ (thin green curve) are plotted versus time in the left panel. The right panel shows the solution trajectory $(x(t),y(t))$ plotted on the phase plane (red curve). If the eigenvalues of $A$ are real, then the right panel also shows the directions of the eigenvectors of $A$ (purple lines). If the corresponding eigenvalue is not zero, arrows along the eigenvector indicates the direction the solution moves along the eigenvector direction.

You can change the initial conditions $\vc{x}_0$ by moving any of the points at the beginning of the solution curves. You can also change the initial condition, the matrix $A$, and the graphing parameters using the control panel at the top. The solution, eigenvalues, and eigenvectors for the chosen parameters are shown in the sections at the bottom of the applet.

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