A Level · Cambridge (CIE) · 9709

Solve separable equations using logarithms partial fractions and inverse tangent

Choose an integration method for each separated side and keep the resulting branch and equilibrium restrictions.

Section 3.8 · Differential equations

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 positive quadratic reciprocals and linear partial fractions.
Use principal inverse-tangent values and apply initial conditions.

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

Quick readiness check

What is ∫dx/(4 + x²)?

What you will learn

Use inverse-tangent and partial-fraction primitives in separated equations.
Retain equilibria and the interval of an explicit or implicit initial-value solution.

Both separated sides can need Paper 3 methods

For dy/dx = (1 + y²)/(1 + x²), separate dy/(1 + y²) = dx/(1 + x²). Integration gives arctan y = arctan x + C in radians. With y(0) = 1, C = π/4. On the interval containing zero where that angle stays in the principal range, the tangent addition formula gives y = (x + 1)/(1 − x), valid for x < 1. At x = 1 the solution has a pole; the formula at x > 1 belongs to a separate branch and does not continue the initial-value solution through it.

A scaled quadratic changes its primitive: ∫dx/(4 + x²) = (1/2)arctan(x/2), not arctan x. Keep the exact multiplier when it appears in a separated equation. An implicit relation such as y + y³/3 = x² + C can be a complete answer when an elementary explicit formula is unhelpful. Differentiate it to check (1 + y²)y′ = 2x rather than treating y as constant.

Partial fractions also expose excluded equilibria

For dy/dx = y(1 − y), division is valid only away from y = 0 and y = 1. Since 1/[y(1 − y)] = 1/y + 1/(1 − y), integration gives ln|y| − ln|1 − y| = x + C, or ln|y/(1 − y)| = x + C. The second logarithm has a minus sign because the derivative of 1 − y is −1. Retain the original constant solutions 0 and 1 separately.

A non-equilibrium branch can be written y = Ke^x/(1 + Ke^x), where the sign of K is retained and the denominator must stay non-zero. For y(0) = 1/4, K = 1/3 and y = e^x/(3 + e^x), defined for every real x and between 0 and 1. Derivative checking distinguishes the correct logistic expression from an answer with the wrong sign in its denominator. Absolute-value logarithms do not justify dropping a negative multiplicative constant on other branches.

Pause and explain

After dividing dy/dx = y(1 − y) by y(1 − y), which constants must be checked separately?

Put the idea to work

Worked example

Solve dy/dx = (1 + y²)/(1 + x²) with y(0) = 1 and state the interval containing the initial point.

Show the worked solution

Integrate the separated reciprocals to get arctan y = arctan x + C.
The initial value gives C = π/4; apply tangent addition to obtain y = (x + 1)/(1 − x).
Its derivative is 2/(1 − x)², equal to the original right-hand side; the initial branch has x < 1.

Answer y = (x + 1)/(1 − x), x < 1 for the branch through x = 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 = (1 + y²)/(4 + x²), y(0) = 0.

Give me a hint

Use the scaled inverse-tangent primitive on the x side.

Compare my reasoning

Integrate dy/(1 + y²) = dx/(4 + x²) to obtain arctan y = (1/2)arctan(x/2) + C.
At x = 0, y = 0, so C = 0.
Thus y = tan[(1/2)arctan(x/2)]; the inner angle lies between −π/4 and π/4 for all real x.

y = tan[(1/2)arctan(x/2)], for all real x.

Look for these in your work

Retain the quadratic scale and its multiplier.
Check the initial value and that the tangent has no pole on the stated interval.

2 · Independent

Solve dy/dx = y(1 − y) with y(0) = 1/4. Also identify the constant solutions of the original equation.

Give me a hint

Use partial fractions and preserve the equilibria excluded by division.

Compare my reasoning

For non-equilibrium branches, ln|y/(1 − y)| = x + C.
The initial ratio is 1/3, giving y = e^x/(3 + e^x), which stays strictly between 0 and 1.
The constant solutions y = 0 and y = 1 also satisfy the original equation but not this initial condition.

y = e^x/(3 + e^x); original equilibria y = 0 and y = 1.

Look for these in your work

Keep the logarithmic minus sign and initial ratio.
Distinguish other solutions of the equation from the particular starting condition.

3 · Transfer

A supplied dimensionless response satisfies dy/dx = 2x/(1 + y²), with y(0) = 1 and x ≥ 0. Find an implicit solution and the first x at which y = 2.

Give me a hint

Integrate (1 + y²) dy = 2x dx and use the starting response.

Compare my reasoning

Integration gives y + y³/3 = x² + C; the initial value gives C = 4/3.
At y = 2 the left side is 14/3, so x² = 10/3.
The first non-negative input is x = √(10/3); the function y + y³/3 is strictly increasing, so it selects one real response at each x.

y + y³/3 = x² + 4/3; reaches y = 2 at x = √(10/3).

Look for these in your work

Differentiate the implicit relation to recover the rate law.
Choose the non-negative input and check that the threshold response is unambiguous.

Common mistakes

Use inverse-trigonometric primitives in radians with their scaling factors.
The primitive of 1/(1 − y) is −ln|1 − y|.
Dividing by y(1 − y) hides both equilibrium solutions.

Recall without your notes

What can you explain now?

What is ∫dy/[y(1 − y)] away from y = 0 and 1?

Compare with the explanation

ln|y| − ln|1 − y| + C.

The partial fractions are 1/y + 1/(1 − y), and the second denominator has derivative −1.

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Make it stick

Identify each separated integrand, integrate with the right scaling and signs, and check the particular branch and any equilibria before interpreting the answer.

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.