Difference Equations

Discrete Change, Sequences, and Differences

Many things in biology, finance, and economics only change at set times — once a year, once a month, or once a day. We number the steps $n = 0, 1, 2, 3, \dots$ and write $u_n$ for the value at step $n$, so $u_0$ is the starting value, $u_1$ is the value after one step, and so on.

Why a Difference Equation?

When an exam question asks why a difference equation is appropriate rather than a differential equation, state two facts:

1. The quantity is recorded and updated at separate, fixed times (such as annually, monthly, or per generation), so the model produces one distinct value for each whole-numbered step.
2. A differential equation would be used instead if the quantity changed continuously and smoothly at every instant, such as a cooling liquid or motion under a variable force.

Sequence Differences

The first difference measures the change between consecutive terms:

Taking the difference of consecutive first differences gives the second difference:

Looking at differences lets us identify the underlying formula of a data table:

• Constant first differences ($\Delta u_n = d$): linear sequence, $u_n = u_0 + nd$.
• Constant second differences ($\Delta^2 u_n = 2a$): quadratic sequence, $u_n = an^2 + bn + c$.
• Constant third differences: cubic sequence ($an^3 + bn^2 + cn + d$).
• Constant ratio ($u_{n+1}/u_n = r$): geometric sequence, $u_n = u_0 r^n$.

For example, consider the sequence 2, 7, 18, 41, 82:

• First differences: $7 - 2 = 5$, $18 - 7 = 11$, $41 - 18 = 23$, $82 - 41 = 41$
• Second differences: $11 - 5 = 6$, $23 - 11 = 12$, $41 - 23 = 18$
• Third differences: $12 - 6 = 6$, $18 - 12 = 6$

Because the third differences are constant ($+6$), the sequence is cubic. We can work forward: the next second difference is $18 + 6 = 24$, the next first difference is $41 + 24 = 65$, so the next term is $u_5 = 82 + 65 = 147$.

Classification and Model Setup

Before solving an equation, you must classify it. In exams, classification marks depend on giving clear reasons rather than just stating labels.

• Order: The difference between the highest and lowest subscripts of the sequence variable. For example, $u_{n+1} = 1.04u_n - 80$ is first order (subscripts differ by 1), whereas $u_{n+2} - 4u_{n+1} + 3u_n = 0$ is second order (subscripts differ by 2).
• Linearity: An equation is linear if every sequence term ($u_n, u_{n+1}, u_{n+2}$) appears only to the first power, terms are not multiplied together, and no sequence term is inside a function like a square root or sine. For example, $u_{n+2} - 3u_{n+1} + 2u_n = n^2$ is linear because $n^2$ depends only on the step number $n$, not on $u$. If an equation contains $u_n^2$ or $u_n u_{n+1}$, it is non-linear.
• Homogeneity: Collect all terms containing the sequence variable on the left-hand side. If the right-hand side is zero, the equation is homogeneous because there is no term independent of $u$. If an isolated constant, an exponential term, or a polynomial in $n$ remains on the right, the equation is inhomogeneous.

Setting Up Timelines and Incremental Models

Always define your timeline before writing an equation: 'Let $P_n$ be the population at the start of year $n$, with $n = 0$ at the start of 2024, so $P_0 = 8000$.'

Follow the chronological order of events described in the problem:

• If interest is added first and then a payment $R$ is deducted: $B_{n+1} = (1 + i)B_n - R$.
• If the payment is deducted at the start of the period before interest is applied: $B_{n+1} = (1 + i)(B_n - R)$.

A common Leaving Certificate formulation involves successive changes: 'the increase on any day is twice the increase on the previous day'. Translate this directly using first differences:

Rearranging into standard form gives $u_{n+2} - 3u_{n+1} + 2u_n = 0$, which is a second-order linear homogeneous equation.

First-Order Linear Difference Equations

The general first-order linear difference equation with constant coefficients is:

where $p$ is the proportional growth or decay multiplier, $q$ is a constant added or subtracted at each step, and $u_0$ is the given initial value.

Homogeneous Form ($q = 0$)

When $q = 0$, $u_{n+1} = p u_n$. This generates a geometric sequence with closed-form solution:

If $p > 1$, the sequence grows without bound. If $0 < p < 1$, it decays towards zero. If $p < 0$, terms alternate in sign.

Inhomogeneous Form ($q \neq 0, p \neq 1$)

When a constant amount is added or removed each step, the sequence may possess an equilibrium value (steady state), denoted $u^*$. Setting $u_{n+1} = u_n = u^*$ gives:

Let $v_n = u_n - u^*$, which measures the distance between the sequence and its equilibrium. Then $v_{n+1} = p v_n$, meaning this gap changes by a factor of $p$ at every step. In terms of $n$, $v_n = (u_0 - u^*)p^n$. Adding $u^*$ back gives the general solution:

This formula is provided in the official Formulae and Tables booklet. You can quote and apply it directly in examinations.

Stability and Long-Term Behaviour

• If $|p| < 1$, $p^n \to 0$ as $n \to \infty$. The sequence approaches $u^* = \frac{q}{1 - p}$, making the equilibrium stable. If $0 < p < 1$, the values move smoothly towards $u^*$. If $-1 < p < 0$, the sequence oscillates around $u^*$ with shrinking swings.
• If $|p| > 1$, $p^n$ grows in magnitude, so the sequence diverges away from $u^*$ (unless $u_0 = u^*$ exactly). The equilibrium is unstable.
• If $p = 1$, the relation becomes $u_{n+1} = u_n + q$, which is an arithmetic sequence solved by $u_n = u_0 + nq$.

Second-Order Homogeneous Difference Equations

A second-order linear homogeneous difference equation with constant real coefficients has the form:

Because each step depends on two previous values, finding a unique solution requires two initial conditions, typically $u_0$ and $u_1$.

Trying a trial geometric solution $u_n = r^n$ leads directly to the characteristic equation:

The structure of the general solution depends on the roots of this quadratic:

Every second-order difference equation examined in this course has a characteristic quadratic that factorises over the real numbers. If your quadratic produces a negative discriminant, stop and check your algebra; an error in signs during rearrangement is the usual cause.

When one root is $r = 1$, the term $A(1)^n$ simplifies simply to the constant $A$. To evaluate the arbitrary constants $A$ and $B$, substitute $n = 0$ and $n = 1$ using your known values $u_0$ and $u_1$, and solve the two simultaneous linear equations.

Second-Order Inhomogeneous Difference Equations

When an external input or function of $n$ appears on the right-hand side, the equation is inhomogeneous:

The complete general solution is made up of two parts:

where $u_n^{(c)}$ is the complementary solution (the solution to the homogeneous equation when the right side is replaced by 0), and $u_n^{(p)}$ is a particular solution satisfying the full equation.

Trial Forms for the Particular Solution

Choose a trial form that matches the family of $f(n)$:

Substitute the trial into the inhomogeneous equation and equate coefficients to find the constants $C, D, \dots$

Order of operations: combine the complementary and particular solutions into $u_n = u_n^{(c)} + u_n^{(p)}$ first, and only then substitute $u_0$ and $u_1$ to find $A$ and $B$. If you evaluate $A$ and $B$ before adding $u_n^{(p)}$, your constants will be incorrect.

Note: This complementary-plus-particular method works for first-order equations too. For example, $u_{n+1} - 3u_n = 8$ has complementary solution $A(3)^n$. Trialling $u_n^{(p)} = C$ gives $C - 3C = 8 \implies C = -4$, so $u_n = A(3)^n - 4$, which matches the equilibrium method.

Numerical Iteration and Spreadsheet Solutions

You do not always need an explicit algebraic formula to solve a difference equation. If you know the starting value, you can generate terms step by step through numerical iteration.

Take the loan repayment recurrence $B_{n+1} = 1.01 B_n - 300$ with starting balance $B_0 = 10000$:

After four payments, the balance is €9,187.92.

Essential Habits for Numerical Work

• Keep full calculator precision throughout intermediate steps, rounding only the final requested answer.
• After finding a closed-form formula, verify it against your table. Using $B_n = 30000 - 20000(1.01)^n$ for $n = 2$ gives $30000 - 20000(1.0201) = 9598$, confirming the algebra in seconds.
• When asked to use digital tools or a spreadsheet, place the step index $n$ in column A and the state variable in column B. Entering =1.01*B2 - 300 and filling down demonstrates the computational procedure.

Non-Linear Difference Equations and Epidemic Models

When interactions between variables depend on capacity limits or contact rates, the equations become non-linear. Characteristic equations cannot solve non-linear equations; we analyse them by determining equilibria algebraically and calculating terms numerically.

Discrete Restricted (Logistic) Growth

When a population experiences resource limitations, growth is constrained by an environmental carrying capacity $K$:

Expanding introduces a $P_n^2$ term, confirming non-linearity. Setting $P_{n+1} = P_n = P^*$ yields the equilibria:

This gives two equilibrium states: extinction at $P^* = 0$, and sustainable capacity at $P^* = K$.

Discrete SIR Epidemic Model

A population of size $N$ is divided into Susceptible ($S$), Infectious ($I$), and Recovered ($R$) individuals across discrete time periods (such as days or weeks):

Here $\beta$ is the transmission parameter and $\gamma$ is the recovery fraction per period. The product $S_n I_n$ reflects the rate of contact between susceptible and infectious individuals. The total population $N = S_n + I_n + R_n$ stays constant at every step because anyone leaving one category enters another ($S_{n+1} + I_{n+1} + R_{n+1} = S_n + I_n + R_n$).

For example, with $N = 1000$, $S_0 = 990$, $I_0 = 10$, $R_0 = 0$, $\beta = 0.0005$, and $\gamma = 0.2$:

• $S_1 = 990 - 0.0005(990)(10) = 990 - 4.95 = 985.05$
• $I_1 = 10 + 4.95 - 0.2(10) = 12.95$
• $R_1 = 0 + 0.2(10) = 2.00$

Notice that $985.05 + 12.95 + 2.00 = 1000$, confirming total population conservation.

Graphical Analysis, Cobweb Diagrams, and Model Evaluation

Two graphical methods are valuable for discrete models:

1. Plot of Terms Versus Time Index

Plot $n$ on the horizontal axis and $u_n$ on the vertical axis as individual points. Do not join them with a continuous line, as the system only exists at whole-number steps. A sequence flattening towards a horizontal line indicates convergence to a stable equilibrium; points alternating above and below a level line reveal oscillation; points curving outward show divergence.

2. Staircase and Cobweb Diagrams

For any first-order relation $u_{n+1} = f(u_n)$, sketch the curve $y = f(u)$ and the diagonal line $y = u$ on the same axes. The intersection where $f(u) = u$ identifies the equilibrium $u^*$.

To trace the sequence from an initial value $u_0$ on the horizontal axis:

1. Move vertically to the curve $y = f(u)$; the vertical height is $u_1$.
2. Move horizontally to the diagonal line $y = u$; this shifts the value $u_1$ onto the horizontal coordinate.
3. Move vertically back to the curve to find $u_2$, and continue the process.

If the curve has a slope between $-1$ and $+1$ at the crossing point, the steps spiral or step inwards towards $u^*$ (a stable equilibrium). If the magnitude of the slope exceeds 1, the path moves away (an unstable equilibrium).

In agricultural economics, current supply depends on the previous season's price ($S_n = c + d P_{n-1}$), while demand depends on current price ($D_n = a - b P_n$). Equating $D_n = S_n$ produces the cobweb model:

Because the slope $-d/b$ is negative, the path forms a rectangular spiral around the equilibrium $P^* = \frac{a - c}{b + d}$. The market converges if $d < b$ (supply is less sensitive to price changes than demand).

Interpreting and Criticising Models

Exam questions routinely ask you to evaluate your model. Address these three areas:

• Meaning: State what the numbers predict in context with units (e.g. 'predicts 8,110 animals after 10 years').
• Assumptions: Identify assumptions made to create the model (such as a constant growth rate, unchanging repayment terms, or a closed population without migration).
• Limitations: Note where the model breaks down. Linear models with $p > 1$ predict infinite growth, which is impossible in a finite environment; a logistic model with a carrying capacity provides a more realistic long-term model.

Key terms

difference equation

An algebraic relation linking the terms of a sequence at discrete intervals, expressing future values in terms of one or more previous values.

order

The difference between the highest and lowest subscripts of the sequence variable appearing in a difference equation.

linear

A difference equation where all sequence terms appear to the first power, are not multiplied together, and do not sit inside non-linear functions.

non-linear

A difference equation containing products, powers, or non-linear functions of sequence terms, such as products or squared terms.

homogeneous

A difference equation where, after collecting all terms containing the sequence variable on one side, the opposite side is zero.

inhomogeneous

A difference equation that contains a non-zero forcing term (constant, polynomial in n, or exponential) independent of the sequence variable.

first difference

The change between consecutive terms of a sequence, given by Δu_n = u_(n+1) - u_n.

second difference

The difference between consecutive first differences, given by Δ²u_n = u_(n+2) - 2u_(n+1) + u_n.

characteristic equation

The quadratic equation a r^2 + b r + c = 0 formed from a second-order linear homogeneous difference equation, whose roots determine the complementary solution.

complementary solution

The general solution to the associated homogeneous equation obtained by setting any external forcing terms to zero.

particular solution

A specific sequence satisfying the complete inhomogeneous difference equation, matching the mathematical form of the forcing term.

equilibrium value

A constant steady-state value u such that if u_n = u, then u_(n+1) = u for all future steps; for u_(n+1) = p u_n + q, u = q / (1 - p).

stable

An equilibrium state that the sequence converges toward over time, which occurs in first-order linear models when the multiplier satisfies |p| < 1.

unstable

An equilibrium state that the sequence moves away from over time, occurring in first-order linear models when |p| > 1.

carrying capacity

The maximum sustainable population size K that an environment can support in a restricted growth model.

cobweb model

An economic recurrence model where supply responds to prior prices and demand responds to current prices, generating stepped or spiralling price paths around equilibrium.