Systems of Differential Equations

You will learn how to convert higher-order differential equations into equivalent systems of first-order equations and solve those systems using eigenvalues and eigenvectors.

Many real-world problems involve multiple interacting quantities. We can rewrite a higher-order equation as a system of first-order equations.

For example, the second-order equation $y'' + 3y' + 2y = 0$ becomes:

$$x_1' = x_2$$ $$x_2' = -2x_1 - 3x_2$$

where $x_1 = y$ and $x_2 = y'$.

In matrix form: $\mathbf{x}' = A\mathbf{x}$, where

$$A = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}$$

The general solution depends on the eigenvalues and eigenvectors of $A$.

The phase portrait shows trajectories in the $x_1$-$x_2$ plane for a stable system. Multiple initial conditions all flow toward the origin along paths determined by the eigenvectors.

Example 1: Converting to a System

Convert $y'' - 5y' + 6y = 0$ into a system.

Solution:

$$x_1' = x_2$$ $$x_2' = -6x_1 + 5x_2$$

Example 2: Solving a System

Solve the system $\mathbf{x}' = A\mathbf{x}$ where $A = \begin{bmatrix} -1 & 1 \\ 0 & -2 \end{bmatrix}$.

Solution: Eigenvalues: $\lambda_1 = -1$, $\lambda_2 = -2$.

Eigenvectors: $\mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$, $\mathbf{v}_2 = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$

General solution:

$$\mathbf{x}(t) = C_1 e^{-t} \begin{bmatrix} 1 \\ 0 \end{bmatrix} + C_2 e^{-2t} \begin{bmatrix} 1 \\ -1 \end{bmatrix}$$

Example 3: With Initial Conditions

Solve with $\mathbf{x}(0) = \begin{bmatrix} 3 \\ -1 \end{bmatrix}$.

Solution: At $t = 0$: $C_1 \begin{bmatrix} 1 \\ 0 \end{bmatrix} + C_2 \begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 3 \\ -1 \end{bmatrix}$

From row 2: $-C_2 = -1 \implies C_2 = 1$. From row 1: $C_1 + 1 = 3 \implies C_1 = 2$.

$$\mathbf{x}(t) = 2e^{-t} \begin{bmatrix} 1 \\ 0 \end{bmatrix} + e^{-2t} \begin{bmatrix} 1 \\ -1 \end{bmatrix}$$

Systems of differential equations are fundamental in modeling coupled oscillators (multiple springs), predator-prey relationships in biology (Lotka-Volterra equations), electrical networks with multiple loops, chemical reaction networks, and even epidemiological models (SIR models for disease spread). In computer graphics and game physics, they simulate realistic multi-body interactions.