A Level · Cambridge (CIE) · 9709

Separate variables and retain solutions lost by division

Integrate a separable equation, keep a general constant and check constant solutions before dividing by a dependent-variable factor.

Pure Mathematics 3 · Differential equations · 3.8

Exam scope and assessment notes

Pure Mathematics 3 (Paper 3, A Level) · Mathematics 9709, 2026–2027, section 3.8. Paper 1 calculus and Paper 3 Integration are assumed. This pathway uses first-order separable equations, including Paper 3 integration methods, with all context assumptions supplied. Second-order equations and integrating-factor methods are beyond this mapped requirement. Check factors before dividing, retain equilibrium solutions, and state the interval and physical meaning of an initial-value solution. Written practice is self-checked; human subject-teacher review remains pending.

Before you begin

Integrate powers, reciprocals and simple composites.
Use logarithms on valid domains and substitute into an original derivative equation.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

What is an antiderivative of 1/y on an interval not containing zero?

What you will learn

Separate a first-order equation into one variable on each side and integrate.
Identify equilibrium solutions and domain restrictions that division can hide.

The three curves are members of y = Ae^(x²), which solves dy/dx = 2xy. Blue has A = 1, green has A = 0 and red has A = −1. At x = 0 the marked starting values are 1, 0 and −1 respectively. The zero member is the equilibrium that dividing by y would initially exclude. Each curve satisfies the same equation; an initial condition selects a member. The plot window is −1 ≤ x ≤ 1, and its horizontal and vertical scales differ. The family itself is defined for every real x.

Separation turns the equation into two integrals

For dy/dx = f(x)g(y), on an interval where g(y) ≠ 0, write dy/g(y) = f(x) dx and integrate both sides. The resulting relation contains one arbitrary constant; separate constants from both integrals can be combined. Keep an implicit relation when solving explicitly would introduce a branch choice. The procedure is justified along a differentiable solution, not by allowing unrelated values of x and y in each step.

For dy/dx = 2xy, non-zero solutions give dy/y = 2x dx, then ln|y| = x² + C. Absorbing the sign and exponential constant gives y = Ae^(x²), with non-zero A on those branches. The original equation also admits y = 0, omitted when dividing by y. Allowing A = 0 in the final family restores it. Every member has derivative 2xy; different values of A describe different starting conditions.

Check what division and square roots exclude

A zero of g(y) may provide a constant equilibrium solution if the original equation is defined there. Test it directly before dividing. For dy/dx = 3xy², non-zero solutions satisfy −1/y = 3x²/2 + C, while y = 0 is another solution. In contrast, dy/dx = x/y is undefined at y = 0, so zero is not a solution of that equation. Multiplying by y gives y² = x² + C, on branches where y stays non-zero.

After y² = x² + C, an explicit branch is y = ±√(x² + C), valid only where the radicand is positive. The choice of sign cannot change through a point where the original equation is undefined. An equation such as dy/dx = x + y is not directly of the separated product form; writing dy/y = x dx from it would discard the addition. Check algebra before integrating and check the result by differentiation afterwards.

Pause and explain

Which equation permits the constant solution y = 0?

Put the idea to work

Worked example

Find the general family for dy/dx = 2xy, including any constant solution lost by division.

Show the worked solution

For y ≠ 0, integrate dy/y = 2x dx to obtain ln|y| = x² + C.
Write the non-zero branches as y = Ae^(x²), with their sign carried by A.
Check y = 0 in the original equation; allowing A = 0 includes it in the family.

Answer y = Ae^(x²), A any real constant, including A = 0.

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Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

Solve dy/dx = 3xy² in general, retaining any equilibrium solution.

Give me a hint

For non-zero y use ∫y⁻² dy; then check the factor you divided by.

Compare my reasoning

Separate dy/y² = 3x dx and integrate −1/y = 3x²/2 + C.
This describes non-zero branches where the denominator of y = −1/(3x²/2 + C) does not vanish.
Substituting y = 0 directly gives zero on both sides, so it must also be included.

−1/y = 3x²/2 + C for non-zero branches; also y = 0.

Look for these in your work

Integrate the reciprocal square with its negative sign.
State both the equilibrium and the non-zero branch restrictions.

2 · Independent

Find an implicit general solution of dy/dx = x/y and explain its real branch restrictions.

Give me a hint

Multiply by y only where the original equation is defined.

Compare my reasoning

Use y dy = x dx, giving y²/2 = x²/2 + C₁.
Combine constants to obtain y² = x² + C.
The explicit branches are ±√(x² + C), on intervals with x² + C > 0; y = 0 is excluded by the original equation.

y² = x² + C; choose a non-zero branch where x² + C > 0.

Look for these in your work

Retain a single arbitrary constant.
Check the original denominator when stating valid branches.

3 · Transfer

A supplied model for a non-negative amount y is dy/dt = −2y. Find its general family and explain the permitted constant and whether it can stay zero.

Give me a hint

Separate for positive y, then test zero in the original model.

Compare my reasoning

For positive y, ln y = −2t + C.
Thus y = Ae^(−2t); non-negative amounts require A ≥ 0.
A = 0 gives y = 0 with zero derivative, so a zero initial amount can remain zero under this law.

y = Ae^(−2t), A ≥ 0 for this non-negative model; A = 0 is valid.

Look for these in your work

Separate the mathematical constant from its physical sign restriction.
Retain the zero amount when interpreting the family.

Common mistakes

Check zero factors before dividing by them.
A denominator in the original equation cannot be set to zero.
One implicit solution can have different explicit branches on separate intervals.

Recall without your notes

What can you explain now?

Why must y = 0 be checked before dividing dy/dx = 2xy by y?

Compare with the explanation

Division by y excludes the valid equilibrium y = 0.

The original equation gives derivative zero at that constant function.

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Make it stick

Separate a product-form equation, integrate with one constant, and check every divided factor against the original equation before accepting a general solution.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.