H2 Maths differential equations formula sheet: separable variable method, given-substitution workflow, general and particular solutions, and Newton's law of cooling modelling -...
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 dxdy=f(x)g(y)
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
Step
Form
Identify separable form
dxdy=f(x),g(y)
Rearrange
g(y)1,dy=f(x),dx
Integrate both sides
∫g(y)1,dy=∫f(x),dx+C
Given-substitution workflow
Step
Action
Let u=h(x,y) as stated in the question
Write the substitution explicitly
Differentiate with respect to x
Obtain dxdu in terms of dxdy
Replace all occurrences
Produce a separable DE in u and x
Integrate, then substitute back
Recover a solution in x and y
General vs particular solution
Solution type
How to obtain
General solution
Integrate and include constant +C
Particular solution
Apply the given initial condition y(x0)=y0 to find C
Modelling: Newton's law of cooling
Quantity
Form
Rate equation
dtdT=−k(T−Tenv), where k>0
General solution
T(t)=Tenv+(T0−Tenv),e−kt
Sign check
If T>Tenv, then dtdT<0
Method Map: Choose the Route Before Integrating
Use this selector at the start of a DE question. Most errors happen before the first integral.
What you see in the question
What to do first
Common trap
dxdy=f(x)g(y)
Move all y-terms with dy and all x-terms with dx.
Integrating g(y) instead of 1/g(y).
"Given u=..."
Differentiate the substitution with respect to the independent variable.
Substituting u but forgetting du/dx.
A modelling statement such as cooling or growth
Define the variables and the sign of the rate first.
Getting a solution that moves away from the stated equilibrium.
A point such as y(0)=2
Save it for after integration and the constant +C.
Applying the point before the general solution is complete.
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: dxdy=f(x)g(y).
Rearrange: g(y)1dy=f(x)dx.
Integrate both sides and include constant of integration.
Example -- Logistic style
Solve dxdy=y(3−y).
Separate: y(3−y)1dy=dx.
Partial fractions: y(3−y)1=31(y1+3−y1).
Integrate: 31(ln∣y∣−ln∣3−y∣)=x+C
Solve for y3−yy=Ae3x⟹y=1+Ae3x3Ae3x
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.
Original DE clue
What to test before dividing
What it means
Common trap
dxdy=y(3−y)
Set y(3−y)=0.
y=0 and y=3 are constant solutions of the original DE.
Dividing by y(3−y) and reporting only the non-constant family.
dxdy=ky
Set y=0.
y=0
dxdy=f(x)g(y)
Solve g(y)=0
Worked check: for dxdy=y(3−y), the functions y=0 and y=3 both give dxdy=0 and make the right-hand side zero. They should be listed alongside the separated non-constant solution unless the initial condition rules them out.
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 dxdy=(x+y)2 given u=x+y, with y(0)=0.
Let u=x+y. Then dxdu=1+dxdy=1+u2.
Separate: 1+u21du=dx.
Integrate: tan−1u=x+C so u=tan(x+C).
Substitute back: y=u−x=tan(x+C)−x.
Apply y(0)=0: tanC=0⇒C=0, hence y=tanx−x
Why the substitution helps: the original right-hand side depends on x+y, not on x and y separately. Naming x+y as u turns the right-hand side into u2, and differentiating u=x+y supplies the missing du/dx link.
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.
Given substitution
Differentiate first
What to replace
Common trap
u=x+y
dxdu=1+dxdy
Replace x+y with u, then replace dxdy using the differentiated line.
Writing dxdu=dxdy and losing the derivative of x
u=y/x
Use y=ux, so dxdy=u+xdxdu
u=ax+by
dxdu=a+bdxdy
Worked check: if u=x+y and the original DE gives dxdy=(x+y)2, then dxdu=1+(x+y)2=1+u2. Now the DE is separable in u and x.
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: dtdT=−k(T−Tenv).
Solve separable equation to obtain T(t)=Tenv+(T0−Tenv)e−kt.
Use given temperature at time t to determine k.
Modelling sign checkpoint
Before solving a modelling DE, test whether the rate you wrote moves the quantity in the direction described by the situation.
Situation cue
First sign check
What the solution should do
Common trap
Cooling toward room temperature
If T>Tenv, then dtdT<0.
T decreases toward Tenv.
Writing a positive rate so the object heats up when it should cool.
Warming toward room temperature
If T<Tenv, then dtdT>0
Population growth proportional to population
If P>0 and growth is stated, then dtdP>0.
P
Decay proportional to amount present
If N>0, then dtdN<0.
N
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.
Check
What to do
Common trap
Differential equation
Differentiate the proposed y, then substitute both y and dxdy into the original DE.
Differentiating correctly but not comparing it with the right-hand side of the DE.
Initial condition
Substitute the given point, such as x=0, into the proposed solution.
Checking the DE but forgetting that a whole family of solutions may satisfy it.
Domain
State where the expression and any division or logarithm used in the solution are valid.
Accepting a solution across a value where the original separation divided by zero.
Worked check: for y=tanx−x, we have dxdy=sec2x−1=tan2x. Since x+y=tanx, the right-hand side of dxdy=(x+y)2 is also tan2x. The condition y(0)=0 also holds because tan0−0=0.
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∣⋅∣ 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) on the right-hand side - instead of rearranging to g(y)1,dy=f(x),dx first - produces a wrong equation. Show the rearrangement step explicitly.
Using ln(y) instead of ln∣y∣: When integrating 1/y, the correct antiderivative is ln∣y∣. Dropping the absolute value signs can cause errors when dealing with domain restrictions or negative values of y
Applying the initial condition before completing the general solution: Substituting the initial condition too early (e.g. before isolating y from the implicit equation) leads to an incorrect particular solution. Always obtain the general solution in full before applying y(x0)=y0.
Forgetting to state the long-term behaviour for modelling questions: Many marks for modelling DEs ask you to interpret the solution physically (e.g. equilibrium value as t→∞). Omitting this interpretation step loses the conclusion marks.
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:
