H2 Maths Differential Equations Formula Sheet

Q: What does H2 Maths Notes (JC 1-2): 5.5) Differential Equations cover?
A: Separable differential equations, given-substitution workflows, and modelling techniques for H2 Maths Topic 5.5.

Download: Get the H2 Maths Differential Equations formula sheet (PDF) for quick revision, or the complete notes (PDF) for the full walkthrough.

Before you revise
Recognise the workflow before you start integrating. A differential equation question usually rewards the setup: identify the variables, separate or use the given substitution, integrate with a constant, apply initial conditions, then verify the final solution.

The core idea is simple: A differential equation links a quantity to its rate of change.

Use it as a working check: First decide whether the equation is already separable or needs the given substitution. Then integrate both sides.

Then go one layer deeper: Initial conditions turn the general solution into one specific model, and verification catches wrong constants or signs.

Concrete example: If temperature changes according to its gap from room temperature, define that gap first. The solution should move toward room temperature, not away from it.

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 5.5 focuses on first-order DEs of the form $\dfrac{\mathrm{d}y}{\mathrm{d}x} = f(x)g(y)$, including making a DE separable using a given substitution.

Formulas at a glance

Every result the 9758 syllabus expects you to apply, on one screen. MF27 does not list any of the solution methods below - you must recall the separation step, the substitution chain, and the particular-solution procedure from memory. Worked examples for each appear in the sections below.

Separable variables

Given-substitution workflow

General vs particular solution

Modelling: Newton's law of cooling

Method Map: Choose the Route Before Integrating

Use this selector at the start of a DE question. Most errors happen before the first integral.

Mini-check before continuing: after separation, each side should be integrable in one variable only. If the left side still mixes $x$ and $y$, the equation has not been separated yet.

First-Order Separable Equations

• Form: $\dfrac{\mathrm{d}y}{\mathrm{d}x} = f(x) g(y)$.
• Rearrange: $\dfrac{1}{g(y)} {\mathrm{d}}y = f(x) {\mathrm{d}}x$

Example -- Logistic style

Solve $\dfrac{\mathrm{d}y}{\mathrm{d}x} = y(3 - y)$.

1. Separate: $\dfrac{1}{y(3 - y)} {\mathrm{d}}y = {\mathrm{d}}x$.
2. Partial fractions: $\dfrac{1}{y(3 - y)} = \dfrac{1}{3}\left( \dfrac{1}{y} + \dfrac{1}{3 - y} \right)$

Common trap: the separation step divides by $y(3-y)$, so values such as $y=0$ or $y=3$ need separate attention if they are possible constant solutions in the original equation.

Constant-solution checkpoint

Before dividing by a factor that contains $y$, check whether setting that factor to zero already gives a constant solution. Separation can hide these equilibrium solutions because division by zero is not allowed.

Worked check: for $\dfrac{\mathrm{d}y}{\mathrm{d}x}=y(3-y)$, the functions $y=0$

Misconception check: an equilibrium solution is not a failed method step. It is a valid solution that can disappear when you divide by the factor that equals zero.

Reducing to Separable Form (Given Substitution)

• Some DEs are not separable in $x$ and $y$ immediately, but the paper may provide a substitution that turns it into a separable first-order equation.
• Workflow: substitute → differentiate in the new variable → separate variables → integrate → substitute back → apply initial condition(s).

Example -- Given substitution

Solve $\dfrac{\mathrm{d}y}{\mathrm{d}x} = (x + y)^2$ given $u = x + y$

1. Let $u = x + y$. Then $\dfrac{\mathrm{d}u}{\mathrm{d}x} = 1 + \dfrac{\mathrm{d}y}{\mathrm{d}x} = 1 + u^2$

Why the substitution helps: the original right-hand side depends on $x+y$, not on $x$ and $y$ separately. Naming $x+y$ as $u$

Given-substitution chain rule checkpoint

When a question gives a substitution, do not replace the expression only. Differentiate the substitution first so the new derivative matches the original DE.

Worked check: if $u = x + y$ and the original DE gives $\dfrac{\mathrm{d}y}{\mathrm{d}x} = (x+y)^2$

Misconception check: a substitution is a change of variables plus a derivative conversion. If the derivative line is missing, the new equation may look simpler but no longer matches the original question.

Modelling with Differential Equations

• Translate physical laws (cooling, population growth, circuits) into DEs.
• Apply initial conditions to determine constants.
• Check units and interpret solutions physically (long-term behaviour, equilibrium).

Example -- Cooling

Newton cooling: $\dfrac{dT}{dt} = -k(T - T_\text{env})$.

1. Solve separable equation to obtain $T(t) = T_\text{env} + (T_0 - T_\text{env}) e^{-kt}$

Modelling sign checkpoint

Before solving a modelling DE, test whether the rate you wrote moves the quantity in the direction described by the situation.

Misconception check: the constant $k$ is usually taken positive. Put the minus sign in the model only when the quantity is decreasing relative to the chosen gap.

Verifying Solutions

• Differentiate proposed solution and substitute back into original equation.
• For initial value problems, check both DE and initial condition(s).
• Discuss domains where solution is valid (avoid division by zero, etc.).

Verification checkpoint

When a question asks you to "verify" a solution, do not just copy the proposed formula into the final line. Show that it passes each condition in the problem.

Worked check: for $y = \tan x - x$, we have $\dfrac{\mathrm{d}y}{\mathrm{d}x} = \sec^2 x - 1 = \tan^2 x$

Misconception check: verification is not the same as solving again. Use the proposed expression as evidence, then show it satisfies the DE, the condition, and the allowed domain.

Calculator Workflow

• GC diffEQ or numerical solver checks solutions; always show analytical steps first.
• Use GC to plot solution behaviour with chosen constants for modelling tasks.
• For exponential solutions, store constants to evaluate quickly at required times.

Exam Watch Points

• Show the separation step clearly (or the given substitution that reduces the DE to separable form).
• Keep constants of integration, then use the given initial condition(s) to get a particular solution.
• Use $\ln\lvert \cdot \rvert$ correctly when integrating rational expressions, and state any domain restrictions if needed.
• If the DE comes from a problem situation, interpret what the solution means (e.g., equilibrium / long-term behaviour).

Practice Quiz

Ensure you can separate variables (and use a given substitution when needed) and interpret modelling constants confidently.

Quick Revision Checklist

• Separate variables and integrate confidently for first-order DEs.
• Follow the given-substitution workflow to reduce a DE to separable form.
• Interpret solutions within modelling contexts, including initial conditions and long-term trends.

Want weekly guided practice on Differential Equations? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.

Common exam mistakes

• Omitting the constant of integration: Forgetting $+ C$ when integrating both sides is one of the most heavily penalised errors. Always write $+ C$ immediately after integrating, then apply the initial condition to find its value.
• Incorrect separation of variables: Writing $dy = f(x)g(y),dx$ and then integrating $g(y)$

Frequently asked questions

Is there a formula sheet for H2 Maths differential equations?
Yes - the "Formulas at a glance" section near the top of this page collects every result you need: the separable-variable procedure, the given-substitution chain, general vs particular solution steps, and the Newton's law of cooling modelling form. Note that MF27 does not include any of these solution methods or the cooling formula - you must recall the separation step, the substitution workflow, and the initial-condition procedure from memory.

Is Topic 5.5 in Paper 1 or Paper 2?
Differential equations are Pure Mathematics and can appear in Paper 1 (100 marks) or Paper 2 Section A (40 marks). Modelling contexts (population, temperature, decay) are particularly common.

Will the exam always give the substitution needed to simplify a DE?
Yes. The 2026 syllabus scope is limited to first-order separable DEs and DEs that can be made separable using a given substitution. You will not need to identify the appropriate substitution yourself - it will be stated in the question.

Can I verify my solution using the GC?
You can use the GC to plot the solution and check behaviour (e.g. long-term equilibrium), but you must show the full analytic method. GC numerical output alone does not earn method marks for integration and initial-condition steps.

Other H2 Maths formula sheets

Revising more than one topic? Grab the matching one-page formula sheet:

• Functions & graphs: Functions · Graphs & Transformations · Equations & Inequalities
• Sequences & series: Sequences & Series
• Vectors: Vectors
• Calculus: Differentiation · Maclaurin Series · Integration Techniques · Definite Integrals · Differential Equations (this page)
• Statistics: Probability · Discrete Random Variables · Normal Distribution · Sampling · Hypothesis Testing · Correlation & Regression

For the official SEAB reference booklet, see the H2 Maths MF27 formula list.

Sources

• SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 5.5 Differential equations (first-order separable $\dfrac{\mathrm{d}y}{\mathrm{d}x} = f(x)g(y)$, including reduction by a given substitution, modelling and initial conditions): https://isomer-user-content.by.gov.sg/334/f27e37f7-f0ec-4a35-b1e8-3a5e88ae2f81/9758_y26_sy.pdf

Next steps: Keep the H2 Maths notes hub open and move into Section 6 - Probability & statistics to link calculus with inference work.