Revise Coupled and second-order differential equations for IB Mathematics: Applications and Interpretation (5.9)

A coupled first-order linear system pairs two rates of change, each depending on both variables. Writing it in matrix form is the first move: it packages the whole system into a single equation and makes the coefficient matrix — the object we will analyse — stand out clearly.

For $\dot{x}=ax+by$ $\dot{y}=cx+dy$ the matrix form is $\dot{\mathbf{x}}=M\mathbf{x}$, where $\dot{\mathbf{x}}=\begin{pmatrix}\dot{x}\\\dot{y}\end{pmatrix}$, $M=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, $\mathbf{x}=\begin{pmatrix}x\\y\end{pmatrix}$.

The point $\mathbf{x}=\mathbf{0}$ (both variables zero) is an equilibrium: if you start there, nothing changes. Everything in this lesson is really a description of how the system behaves near that equilibrium — whether it is drawn in, pushed out, or circles around.

An eigenvector $\mathbf{v}$ of $M$ is a direction that $M$ merely stretches: $M\mathbf{v}=\lambda\mathbf{v}$, with the stretch factor $\lambda$ the eigenvalue. Along such a direction the coupled system decouples into a single exponential, $e^{\lambda t}\mathbf{v}$, because differentiating gives $\lambda e^{\lambda t}\mathbf{v}=M(e^{\lambda t}\mathbf{v})$. For two real distinct eigenvalues the general solution is a linear combination of these two clean exponential solutions.

Eigenvalues: solve $\det(M-\lambda I)=0$.

Eigenvectors: for each $\lambda$, solve $(M-\lambda I)\mathbf{v}=\mathbf{0}$.

General solution (real distinct eigenvalues): $\mathbf{x}=Ae^{\lambda_1 t}\mathbf{v}_1+Be^{\lambda_2 t}\mathbf{v}_2$.

You often do not need the full solution to describe long-term behaviour: the eigenvalues alone fix the shape of the phase portrait — the picture of trajectories in the $xy$-plane. In the mandated example above the eigenvalues $3$ and $-1$ have opposite signs, so the origin is a saddle: trajectories are drawn in along the $\mathbf{v}_2$ direction (the decaying $e^{-t}$ term) but flung out along the $\mathbf{v}_1$ direction (the growing $e^{3t}$ term), making it unstable overall.

A single second-order linear equation can always be recast as a coupled first-order system, which lets you use the eigenvalue method and Euler's method on it. The trick is to name the first derivative as a new variable. For $\ddot{x}+p\dot{x}+qx=0$, set $y=\dot{x}$; then $\dot{y}=\ddot{x}=-p\dot{x}-qx=-qx-py$. Two first-order equations replace one second-order equation.

$\ddot{x}+p\dot{x}+qx=0$, with $y=\dot{x}$, becomes $\dot{x}=y$ $\dot{y}=-qx-py$ i.e. $\dot{\mathbf{x}}=\begin{pmatrix}0&1\\-q&-p\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}$.

When a system has no neat closed form — or you simply want a numerical trajectory — Euler's method steps it forward in small increments. It is the single-variable method run on every component at once, with one golden rule: use the OLD values on the right-hand side to update ALL components before moving on.

For $\dot{x}=f(x,y)$, $\dot{y}=g(x,y)$ with step $h$: $x_{n+1}=x_n+h\,f(x_n,y_n)$ $y_{n+1}=y_n+h\,g(x_n,y_n)$ (both use $x_n,y_n$ — the values before this step).

The same machinery models very different situations. In a predator-prey model, $\dot{x}=ax-by$, $\dot{y}=cx-dy$ couples prey $x$ and predator $y$; complex eigenvalues then predict the oscillating boom-and-bust cycle of the two populations. In coupled tanks, salt concentrations $x$ and $y$ flow between two connected tanks, and negative eigenvalues show both settling to a shared equilibrium. In simple harmonic motion, $\ddot{x}=-\omega^2 x$ becomes $\dot{x}=y$, $\dot{y}=-\omega^2 x$, whose eigenvalues $\pm\omega i$ are purely imaginary — a centre, matching undamped oscillation forever. Reading the eigenvalues tells you which of these stories a given matrix is telling before you compute a single trajectory.

In an exam the marks are analytic: each is tied to a specific line of working. A method mark (M) rewards a correct approach even if a number later slips; an accuracy mark (A) rewards a correct result and is usually dependent on the method mark being earned. Follow-through (FT) means a correct final step performed on your own earlier (wrong) eigenvalue still scores; equivalent correct forms — scaled eigenvectors, differently named constants, vector or component layout — are accepted. All of that protection exists only if your method is on the page. Re-read the mandated worked example above: six marks, each earned by one specific line, three of them method marks that survive an arithmetic slip and two accuracy marks shielded by follow-through onto your own eigenvalues and eigenvectors. A student who writes only the final vector with no characteristic equation, no eigenvectors shown, risks losing four or five marks if a single number is off; a student who shows $\det(M-\lambda I)=0$, the eigenvalues, each $(M-\lambda I)\mathbf{v}=\mathbf{0}$ solved, and the assembled solution keeps the method marks regardless.

Your GDC returns eigenvalues and eigenvectors of a matrix directly — enter $M$ and read them off, remembering that the machine's eigenvectors are often normalised or scaled, so match them to a tidy form by the ratio of components. Use the phase-plane plotter to confirm the nature of an equilibrium: a node has trajectories flowing straight in or out, a saddle has the tell-tale in-one-way-out-the-other pattern, and a spiral winds around the origin. For numerical work, Euler's method is quick to tabulate, but a smaller step $h$ gives a better estimate — halve it if you need more accuracy.

One idea runs through the whole lesson: the eigenvalues and eigenvectors of $M$ decode the coupled system $\dot{\mathbf{x}}=M\mathbf{x}$ — the eigenvectors give the clean directions, the eigenvalues give the growth, decay or rotation, and their signs alone name the equilibrium. Converting a second-order equation to a system, and stepping a system forward with Euler's method, are the same idea reached from two other directions. These tools power the modelling questions that dominate Paper 2 — populations, mixing, oscillation — so master the habit: write $\dot{\mathbf{x}}=M\mathbf{x}$, find the eigenvalues, assemble the solution, and read off the long-term story.