Systems of ODEs Track Interacting Quantities

Real systems rarely contain one changing quantity in isolation. Predator and prey populations interact, chemical species react, and position and velocity evolve together. A system of ODEs tracks these coupled states simultaneously.

Vector form

A first-order system can be written as

$$\mathbf x'=\mathbf f(t,\mathbf x).$$

For a linear autonomous system,

$$\mathbf x'=A\mathbf x,$$

the matrix $A$ describes how every state variable influences every rate of change.

Converting higher-order equations

The second-order equation $y''+cy'+ky=0$ becomes a first-order system by setting $x_1=y$ and $x_2=y'$:

$$\begin{aligned} x_1'&=x_2,\\ x_2'&=-kx_1-cx_2. \end{aligned}$$

This conversion lets one framework and one family of numerical methods handle equations of any order.

Phase planes and equilibria

For a two-state autonomous system, the phase plane plots $x_2$ against $x_1$ rather than either variable against time. Equilibria satisfy $\mathbf f(\mathbf x^*)=\mathbf0$. Trajectories reveal whether nearby states approach, depart from, or circulate around them.

Negative real parts of all eigenvalues indicate decay toward the equilibrium. A positive real part creates an unstable direction. Complex eigenvalues introduce rotation or oscillation.

Nonlinear systems

Near an equilibrium, the Jacobian matrix of $\mathbf f$ provides a linear approximation. Its eigenvalues often predict local behavior, though zero real parts require deeper analysis and distant trajectories may behave differently.

Solve along eigenvector directions

If $A\mathbf v=\lambda\mathbf v$, then $\mathbf x=e^{\lambda t}\mathbf v$ satisfies $\mathbf x'=A\mathbf x$. Independent eigenvectors therefore build the general solution as a linear combination of modes.

Check your understanding

Why can a second-order scalar ODE always be written as two first-order equations?

Show the reasoning

Introduce state variables for the quantity and its first derivative. Then one equation defines the derivative of position as velocity, and the original ODE defines the derivative of velocity.