Systems of Differential Equations

Main idea

When several quantities change together and each one's rate depends on the others, you get a system. A linear system fits in one line, $\vec x' = A\vec x$ , and even a single higher-order equation can be rewritten as one, so it's a single framework for all of them.

\vec x\,' = {\htmlData{sym=0}{A}}\,\vec x, \qquad \vec x = \begin{pmatrix} x_1\\ x_2\end{pmatrix}

What the symbols mean

$A$
Coefficient matrix
Row i says how each quantity drives the rate of x_i.

Explanation

A system of differential equations is simply several equations solved together, because the quantities keep affecting each other: more rabbits means more food for foxes, and more foxes means fewer rabbits. It's good for any model where you can't track one thing alone. The equation $\vec x\,' = A\,\vec x$ looks compact and a bit cryptic, but it's just three pieces: $\vec x$ (a list of all the quantities you're tracking), $\vec x\,'$ (how fast each one is changing), and $A$ (a table of numbers saying how much each quantity pushes on each rate). It's the same kind of equation as $y' = ky$ , just with several quantities at once. Bear with the vector notation; it only saves writing.

Why systems? Predators and prey, infected and susceptible people, currents in two loops of a circuit, two masses joined by springs: real models track several quantities at once.

From one equation to a system. Here's a trick that connects this to everything you've done so far. For $y'' + 3y' + 2y = 0$ , let $x_1 = y$ and $x_2 = y'$ . Then

x_1' = x_2, \qquad x_2' = -2x_1 - 3x_2,

so $A = \begin{pmatrix} 0 & 1\\ -2 & -3\end{pmatrix}$ . The first equation is just the definition ( $y$ 's rate is $y'$ ), and the second is the original equation solved for $y''$ . An $n$ th-order equation becomes $n$ first-order ones this way, so second-order equations are really a special case of systems.

Solutions are vector functions $\vec x(t)$ . For a single equation $y' = ky$ the answer was $e^{kt}$ , so it makes sense to try $\vec x = e^{\lambda t}\vec v$ here. Plugging in gives $\lambda e^{\lambda t}\vec v = Ae^{\lambda t}\vec v$ , which is $A\vec v = \lambda\vec v$ . That leads straight to eigenvalues, which is where the next lesson goes.

A common mistake. It's tempting to think every system with two quantities can be written as $\vec x' = A\vec x$ , because the predator–prey equations also have two rates. It isn't so: $\vec x' = A\vec x$ only covers linear systems, where each rate is a sum of constant multiples of the quantities. Terms like $xy$ in predator–prey make it nonlinear, and we'll study those near their equilibria.

See it

This is the direction field for rabbits x and foxes y, with x′ = x(2 − y) and y′ = y(x − 3). Each arrow shows where the two populations head next. Notice how the arrows circle the point (3, 2), where both populations hold steady: more rabbits feed more foxes, the foxes eat the rabbits, and then the foxes starve, over and over.

Worked examples

Example 1

Write $y'' + 4y = 0$ as a system.

1
Let $x_1 = y$ and $x_2 = y'$ . Then $x_1' = x_2$ by definition, and solving the equation for $y''$ gives $x_2' = -4x_1$ .

$A = \begin{pmatrix} 0 & 1\\ -4 & 0\end{pmatrix}$

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Give the matrix $A$ for $x_1' = x_2$ , $x_2' = -2x_1 - 3x_2$ , row by row.

Problem 2+5 XP

Give $A$ for $y'' + 4y = 0$ written as a system, row by row.

Problem 3+5 XP

Is $\vec x = e^{2t}\begin{pmatrix}1\\1\end{pmatrix}$ a solution of $\vec x' = \begin{pmatrix}1 & 1\\ 2 & 0\end{pmatrix}\vec x$ ?

Problem 4+5 XP

Is the Lotka–Volterra predator–prey system linear?

Problem 5+5 XP

How many first-order equations does a third-order equation become?

Problem 6+5 XP

A spring obeys $2u'' + 3u' + 8u = 0$ . Write it as a first order system $\vec x\,' = A\vec x$ with $x_1 = u$ and $x_2 = u'$ . Give $A$ row by row.

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Write $y''' - 2y'' + y' - 5y = 0$ as $\vec x\,' = A\vec x$ with $x_1 = y$ , $x_2 = y'$ , $x_3 = y''$ . Give the $3\times3$ matrix $A$ row by row.

Where it's used

Every serious simulation package (circuits, chemistry, flight dynamics) works with first-order systems, converting higher-order equations first.

Predator and PreyDiff Eq · Biology Epidemic CurvesDiff Eq · Medicine A Pill's Path from Gut to BloodDiff Eq · Medicine

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