Systems of Differential Equations

Many problems involve several unknown functions that influence one another, so their equations cannot be solved one at a time. Written in matrix form, such a system becomes \mathbf{y}' = \mathbf{A}\mathbf{y}, and the entire solution theory reduces to the eigenvalues and eigenvectors of $𝐀$ . This is the point where linear algebra becomes indispensable to differential equations.

What This Chapter Covers

Section	What you will learn
5.1 Constant Coefficient Systems	Writing a system as \mathbf{y}' = \mathbf{A}\mathbf{y}, and why $𝐲 = e^{λ t} 𝐯$ forces $𝐯$ to be an eigenvector. Includes complex eigenvalues.
5.2 Defective Coefficient Matrices	What to do when $𝐀$ has fewer than $n$ independent eigenvectors, using generalized eigenvectors.
5.3 Nonhomogeneous Systems: Undetermined Coefficients	Guessing $𝐲_{p}$ from the shape of $𝐠 (t)$ , with the modification rules that differ from the scalar case.
5.4 Nonhomogeneous Systems: Variation of Parameters	The fundamental matrix $𝐘 (t)$ and the compact formula \mathbf{u}' = \mathbf{Y}^{-1}\mathbf{g}.

The Central Idea

Substituting the trial solution $𝐲 = e^{λ t} 𝐯$ into \mathbf{y}' = \mathbf{A}\mathbf{y} gives

λ e^{λ t} 𝐯 = 𝐀 e^{λ t} 𝐯 ⟹ 𝐀 𝐯 = λ 𝐯 .

So a nontrivial solution of that form exists precisely when $λ$ is an eigenvalue and $𝐯$ an eigenvector of $𝐀$ . Solving the system becomes the algebraic problem of finding eigenvalues and eigenvectors.

This is the exact analogue of the characteristic equation for a single equation, and the analogy runs deep:

Single equation ay'' + by' + cy = 0	System \mathbf{y}' = \mathbf{A}\mathbf{y}
Trial $y = e^{r x}$	Trial $𝐲 = e^{λ t} 𝐯$
Characteristic polynomial $a r^{2} + b r + c$	Characteristic polynomial $det (𝐀 - λ 𝐈)$
Distinct roots give $e^{r_{1} x}$ , $e^{r_{2} x}$	Independent eigenvectors give $e^{λ_{1} t} 𝐯^{(1)}$ , $e^{λ_{2} t} 𝐯^{(2)}$
Repeated root gives an extra $x e^{r x}$	Defective matrix gives an extra $e^{λ t} (𝐯 t + 𝐮)$
Complex roots give $e^{α x} \cos β x$ and $e^{α x} \sin β x$	Complex eigenvalues give the real and imaginary parts of $e^{λ t} 𝐯$

Systems and Single Equations Are Interchangeable

Any $n$ th order equation can be rewritten as a system of $n$ first-order equations. Given

y^{(n)} = f\left(t, y, y', \dots, y^{(n-1)}\right) ,

set $x_{1} = y$ , x_2 = y', through $x_{n} = y^{(n - 1)}$ . Then

x_1' = x_2, \quad x_2' = x_3, \quad \dots, \quad x_{n-1}' = x_n, \quad x_n' = f\left(t, x_1, \dots, x_n\right) .

Convert y'' + 3y' + 2y = 0 into a first-order system and check that the eigenvalues match the characteristic roots.

Solution

Set $x_{1} = y$ and x_2 = y'. Then x_1' = x_2 and x_2' = y'' = -3y' - 2y = -2x_1 - 3x_2, so

\mathbf{x}' = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}\mathbf{x} .

The characteristic polynomial is

det [\begin{matrix} - λ & 1 \\ - 2 & - 3 - λ \end{matrix}] = λ^{2} + 3 λ + 2,

identical to the characteristic polynomial $r^{2} + 3 r + 2$ of the original equation. The eigenvalues $- 1$ and $- 2$ are exactly the characteristic roots.

This equivalence matters in both directions. It means the systems theory subsumes everything in Chapter 4, and it means numerical software that solves first-order systems can handle equations of any order after this rewriting. It is the standard preprocessing step for numerical work.

Where Systems Appear

Coupled oscillators. Two masses joined by springs exchange energy, and neither displacement can be solved for alone.
Compartment and mixing models. Two tanks that pump fluid into each other give a system for the two concentrations.
Predator and prey. The Lotka-Volterra equations are a nonlinear system whose behavior near equilibrium is studied by linearizing to \mathbf{y}' = \mathbf{A}\mathbf{y}.
Electrical networks. A circuit with several loops yields one equation per loop current.
Compartmental models in medicine. Drug concentrations across blood, tissue, and organs follow a linear system.

Frequently Asked Questions

Why write the system in matrix form?

Because the matrix form exposes the structure. Written out as $n$ scalar equations, a system looks like a mess of interlocking relations. Written as \mathbf{y}' = \mathbf{A}\mathbf{y}, it looks exactly like the scalar equation y' = ay whose solution is $y = c e^{a t}$ , and that resemblance is not superficial: the solution really is $𝐲 = e^{𝐀 t} 𝐜$ , built from eigenvalues and eigenvectors.

How many arbitrary constants does the general solution have?

Exactly $n$ for a system of $n$ first-order equations, one for each independent solution vector. The initial condition is then a vector $𝐲 (t_{0}) = 𝐲_{0}$ with $n$ components, which determines all $n$ constants.

What if the matrix has complex eigenvalues?

They come in conjugate pairs when $𝐀$ is real, and the pair contributes two real solutions: the real and imaginary parts of $e^{λ t} 𝐯$ . The result oscillates, with $e^{α t}$ setting the growth or decay and $β$ setting the frequency. Section 5.1 works this out.

Does every matrix have enough eigenvectors?

No. A matrix with fewer than $n$ independent eigenvectors is called defective, and this can only happen when eigenvalues repeat. Section 5.2 shows how generalized eigenvectors supply the missing solutions, in a way that closely parallels the extra factor of $x$ for repeated roots in Chapter 4.