Deep Dive: Separable Differential Equations

Provenance: written 2026-09-29. Read/checked this session: research/veridian/WMA14-verified-facts.md §1 (spec text for 5.2 and 6.4, verbatim), §3 (formula-booklet contents — confirmed no discrete formula for either spec point), §4 (mark-scheme conventions, M/A/B/ag/ft/cao definitions and the generic differentiation/integration M1 gate), §5.5 (the Jan 2024 Q4 cone citation, spec 5.1/5.2 boundary), §5.6 (the topic's own five citations, spec 6.4), §7–§8 (evidence-strength notes) in full; the existing lesson file content/veridian/curriculum/wma14/separable-differential-equations.ts in full (mechanism derivation, both worked chains, both marked solutions, all six trap-taxonomy items, and its own header/footer provenance notes); content/veridian/technique-hub/wma14/codex-formula-and-method-bank.md §2.8 (the existing seven-line method skeleton) and its §5–6 formula listings; forge-exam-technique-guide.md (its per-question mark table and cross-topic "show that" integrity section); signal-frequency-and-boundaries.md §1 (the topic's own frequency row: 4 of 7 series); content/veridian/question-bank/wma14.ts (checked — contains no item for either spec point yet). No new primary Pearson document was fetched this session; every claim below traces to the facts bank's own already-verified citations, re-synthesized at a level of theoretical and taxonomic depth the lesson file, method skeleton and frequency row do not individually give it. Not affiliated with or endorsed by Pearson Edexcel.

#WMA14 · CODEX Deep-Dive — Differential Equations with Separable Variables, and Forming Them from Rates of Change

#Why this document exists, and what it does not repeat

The lesson file drills this topic through two worked chains, two full marked solutions (one VERIDIAN-original tank problem, one real January 2026 past-paper question), a chain-drill, six trap-taxonomy items and four MCQs. The CODEX formula bank's §2.8 compresses the whole solving procedure into a seven-line method skeleton. Neither is the right place for what this document does instead: laying out the complete real model of what makes a first-order equation separable and why the solving recipe is legal, then taxonomising the exact sub-cases the real WMA14 record actually exercises — sorted by which antiderivative family each side of a separated equation belongs to, not by which worked example happens to illustrate it — and closing with a synthesis of every real exam angle this specific pair of spec points has actually been tested from. This document does not re-run the lesson's own worked chains, its two full marked solutions, or its trap-taxonomy prose; it cites them by name and question and adds the connective layer between them.

Scope, confirmed against the spec, not assumed. Two spec points are genuinely load-bearing here, both verified verbatim against the spec PDF (Specification Issue 3, April 2019, pp. 26–29): 5.2, "Formation of simple differential equations. Guidance: Questions involving connected rates of change may be set," and 6.4, "Analytical solution of simple first order differential equations with separable variables. Guidance: General and particular solutions will be required." The lesson file bundles both under one title because the real exam record does the same — a rates-of-change context is how this equation usually arrives on the page in the first place — and this document follows that same bundling, holding the same boundary the lesson file holds: nothing below claims a citation for second-order equations, non-separable first-order equations, or any technique beyond what these two exact spec points name.

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#1. The theoretical model: what "separable" actually licenses, and what it costs

A first-order differential equation relates a function to its own first derivative only: $\frac{dy}{dx} = (\text{something in } x \text{ and/or } y)$, with no $\frac{d^2y}{dx^2}$ or higher term anywhere. P4 restricts to exactly one solvable sub-type of this: the right-hand side factorises into a function of $x$ alone times a function of $y$ alone, $\frac{dy}{dx} = f(x)g(y)$. This is a genuine restriction, not a simplification of notation — $\frac{dy}{dx} = x+y$ is first order but admits no such factorisation, and needs a different technique entirely, one that has no citation anywhere in this course's WMA14 review and is correspondingly absent below (see §6).

The recipe — separate, integrate each side, apply a boundary condition — looks like it treats $dy$ and $dx$ as ordinary numbers that can be shuffled across an equals sign, which formally they are not. The lesson file's own mechanism block derives, in full, why the shuffle is legal anyway: letting $G(y)$ be any antiderivative of $\frac{1}{g(y)}$, the chain rule turns $\frac{d}{dx}[G(y)]$ into exactly $\frac{1}{g(y)}\cdot\frac{dy}{dx}$, which the original equation lets you rewrite as the plain function $f(x)$ — so integrating both sides with respect to $x$ in the completely ordinary sense reconstructs the "separate and integrate" recipe as a consequence, not a rule pattern-matched onto the notation. That full derivation is not repeated here; what is worth adding at this theoretical layer is what it implies about the constant count: the derivation shows the arbitrary constant enters on only one side of the argument (the $x$-side), which is the real reason a first-order separable equation needs exactly one constant rather than two — not a convention, a direct consequence of where integration actually happens in the proof.

This is also the precise sense in which general and particular solutions differ, and spec 6.4's own guidance makes both explicitly examinable: "General and particular solutions will be required." The general solution — one arbitrary constant left unresolved — names an entire one-parameter family of curves, every member of which genuinely satisfies the original differential equation (the lesson file's own diagram block shows two members of one such family side by side, converging toward the same equilibrium from opposite starting depths, to make this concrete). A boundary condition — one given $(x,y)$ pair the curve must pass through — substitutes into the general solution and pins the constant to one specific value, which is what turns "a whole family" into "exactly one curve." Because the equation is only first order, one $(x,y)$ pair is exactly enough information to do this: there is one constant to pin down, and one equation (the boundary condition) to pin it with — no more information is needed, and no less will do.

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#2. Solving a separable equation (spec 6.4) — the real sub-cases, sorted by antiderivative family

Every separable equation on this spec reduces, after separating, to two ordinary integrals — one in $x$, one in $y$ (or whichever two variables the context uses). The mechanical difficulty of the topic is not the separation step in the abstract; it is which integral family each side turns out to belong to, and the real citation record splits cleanly into two such families, confirmed by two structurally different real questions.

#2.1 Pure reverse-power-rule on both sides — no logarithms anywhere

Real anchor: January 2026, Q3. The equation $\frac{dh}{dt}=\frac{2h^{3/2}}{5t^2}$ separates to $\int\frac{1}{2}h^{-3/2}\,dh=\int\frac15t^{-2}\,dt$ — every exponent on both sides is a plain rational power (one of them negative and fractional), and every antiderivative on both sides is another power of the same variable, never a logarithm. The genuinely distinct mechanical risk this sub-case carries is handling a negative, fractional exponent correctly under the power rule ($h^{-3/2}\to -2h^{-1/2}$) — the same underlying rational-exponent fluency this course's own binomial-expansion-for- rational-$n$ lesson (spec 4.1) drills for a different reason, confirming this is a real, recurring P4 demand on manipulating $x^{p/q}$ rather than a one-off feature of this single question.

Partial second anchor: October 2021, Q2. The examiner's report quotes a documented wrong intermediate step — candidates who "moved the 4" incorrectly arrived at $\int 4/y^2\,dy = \int 1/(4x+5)^{3/2}\,dx$. Whatever the exact placement of the constant 4 in the genuinely correct separation, the quoted fragment confirms both sides of this real question are power-type too: a $y^{-2}$ term on one side, a fractional power of a linear expression in $x$ on the other — which is a second, structurally distinct member of this same family, because integrating $(4x+5)^{-3/2}$ needs the linear-inside reverse-chain-rule scaling factor ($\frac1a$, where $a$ is the coefficient of $x$ inside the bracket) on top of the power rule itself. That scaling-factor slip is independently confirmed, on different questions under spec 6.3, as "the single most-repeated confirmed recurring slip in this entire facts bank" (Oct 2022 Q2; Jan 2024 Q2b) — this document flags the transfer risk explicitly rather than asserting it as a confirmed spec-6.4 citation: no examiner report reviewed for spec 6.4 itself names this exact slip on this exact question, so it is stated here as a real, named, checkable risk inherited from a fluency this paper tests elsewhere, not as a fifth confirmed 6.4 citation.

#2.2 Logarithmic integration — a reciprocal-of-linear (or reciprocal-of-$y$) term on at least one side

Real anchor: January 2021, Q10. The examiner's report states plainly that candidates who found the constant "had difficulty dealing with the logarithms and rearranging" — an explicit, direct confirmation that this real question's solution genuinely involves logarithms, structurally different from Jan 2026 Q3's pure-power case. No full equation is recoverable from the facts bank for this question, so this document states only what is confirmed (logs appear, and rearranging them is where marks are lost), not an invented equation to match.

A mirror-image trap on the same family: October 2021, Q9. This report documents the opposite mistake — candidates "integrating 1/320 to ln(320t)", i.e. inventing a logarithm on a side of the equation where the correct antiderivative of a plain constant with respect to time is simply a linear term, (constant)$\times t$. Read together, Jan 2021 Q10 and Oct 2021 Q9 document two real, opposite-facing failure modes on the same antiderivative family: mishandling logarithms that genuinely belong in the solution, and manufacturing a logarithm where the correct integral has none. Both are confirmed on real WMA14 questions; neither is hypothetical.

#2.3 Separation-step algebra — before any integration has actually started

The same October 2021 Q2 report locates a distinct, earlier failure point: "moved the 4" describes an error in the rearrangement that produces the two integrals in the first place, not in evaluating them. This is confirmed as a genuine, separate risk from either antiderivative family above — a candidate who can integrate both families correctly can still lose the method mark by mis-placing a constant while separating, before either integral is even written down.

#2.4 Final rearrangement algebra — after both sides are correctly integrated

Two distinct real errors are confirmed at this later stage, both algebraic rather than calculus-based. October 2021, Q2 again: after both sides are integrated to a form like $a/y = b\sqrt{4x+5}+c$, converting it to $y = 1/(a\cdot b\sqrt{4x+5}) + 1/c$ distributes a reciprocal across a sum — a genuine algebra error, since $\frac{1}{p+q}\neq\frac1p+\frac1q$, and one wholly independent of whether the integration itself was correct. January 2026, Q3(a)'s real mark scheme confirms a second, distinct rearrangement sub-skill at the same stage of a different question: reaching $h$ from $-h^{-1/2}=-\frac15t^{-1}-\frac45$ correctly requires combining the right-hand side into a single fraction before inverting — not inverting term-by-term — and then squaring both full sides of the resulting equation, not each term of it separately. The mark scheme explicitly condones an arithmetic slip in that squaring step (landing on $5t^2$ instead of $25t^2$) provided the method — squaring a whole side, not a sum of squared terms — is genuinely the one used. Both 2.4 errors sit at the identical point in the procedure (isolating the target variable after correct integration) but are different mistakes: distributing a reciprocal over a sum, versus inverting or squaring term-by-term instead of as a whole expression.

#2.5 General-and-particular integrity, and the "show that" trap

Spec 6.4's own guidance requires both a general and a particular solution, and Jan 2021 Q10 confirms this is genuinely tested (finding the constant, then handling the resulting logarithms). The same report adds a distinct integrity finding for any question where the particular solution is a printed "show that" target: "persuaded many candidates to 'adjust' their working following obvious mistakes." This is a mark-scheme-relevant finding, not a study-skills aside — an "ag" (answer given) line is scored on the genuineness of the derivation reaching it, and a script that quietly changes an intermediate number to land on the printed target is a real, examiner-documented pattern that is specifically not credited even when the final line matches. The lesson file's own tank marked-solution builds its warrantCheck directly around this exact quote; this document names it as its own sub-case because it recurs as a real risk on any ag part of this topic, not only the one worked example that drills it.

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#3. Forming the equation first — connected rates of change (spec 5.2)

Almost no real WMA14 question on this topic hands over $\frac{dy}{dx}=\ldots$ ready-made. A context describes how one quantity changes with time, and the equation that actually gets solved has to be built via the chain rule before separation is even possible: $\frac{dV}{dt}=\frac{dV}{dr}\times\frac{dr}{dt}$, or the equivalent for whichever two quantities the scenario connects. Four real, distinct failure points are confirmed for this setup stage, all before any separation or integration begins.

Choosing the correct geometric formula. October 2021, Q9's report documents candidates "using the volume of the cylinder as V = (4/3)πr²h" — a sphere's volume formula, misapplied to a cylinder, and even mistyped with an $h$ the sphere formula never has. This is not a lookup failure in the ordinary sense: the facts bank's own §3 confirms the exam formula booklet's P1 mensuration entry gives only a sphere's surface area and a cone's curved surface area — no solid's volume, for any shape, appears in the booklet anywhere. Every volume formula this topic needs has to be recalled correctly from memory, cued only by the shape the question actually describes.

Substituting a genuinely varying quantity, instead of holding it fixed. January 2024, Q4's report — a cone question tested via surface area rather than volume, so a related-rates differentiation problem rather than a full equation taken through to integration, and this document holds that scope boundary exactly as the lesson file does — records "treating l in the given S formula as a constant", where the slant height $l$ was actually a function of the radius via Pythagoras and needed substituting in before differentiating. The same report states the question was "not well understood by most" — the strongest single difficulty rating quoted anywhere in this facts bank for this pair of spec points. The general principle this exposes: some geometric quantities in a formula genuinely are constants with respect to the differentiation variable (a cylinder's cross-sectional radius, walls vertical), and some only look fixed but are themselves functions of it — and the exam record shows candidates cannot yet reliably tell the two apart on sight.

Not dropping half a two-term rate. The same Oct 2021 Q9 report documents candidates "using either dV/dt = 0.6π or dV/dt = −0.15πh" — each is exactly half of a genuine two-term rate (an inflow and an outflow), with the other term silently discarded. This is a setup error with a built-in self-check: because the target of that question's own "show that" part is printed, a one-term rate simply will not reach it — provided the discrepancy is chased back to its real cause rather than adjusted away (§2.5's trap, recurring here at the setup stage rather than the integration stage).

Reporting the final answer in the unit the question was actually worked in. The same report's closing line: a numerically correct answer of 208 lost marks for "giving the units for the answer 208 as seconds rather than minutes." Every quantity in that question had been defined in minutes from its first line — the error is not arithmetic, it is a final-step unit mismatch against the question's own stated terms.

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#4. The one genuinely new closing angle: reading a limit off a closed rational solution

Every separable-DE citation on this spec prior to January 2026 stops at finding a particular solution from a boundary condition. January 2026, Q3(b) — the first genuinely real, cited past-paper question this course has for this topic, independently re-verified against both the primary question paper and mark scheme PDFs and re-derived from scratch — adds a real, structurally new second half: "hence find... the limit of the height" as $t\to\infty$, from the closed form $h=\frac{25t^2}{(1+4t)^2}$ already found in part (a). The real mark scheme's method is algebraic, not calculus-based: divide every term by $t^2$ (equivalently, read off the ratio of the two leading $t^2$ coefficients), so that every lower-order term vanishes as $t\to\infty$ and only $\frac{25}{16}$ survives.

This matters as a genuine sub-case, not a restatement of ordinary limit-taking, because the tempting wrong method is independently confirmed to be a dead end for this exact question: $\frac{dh}{dt}= \frac{2h^{3/2}}{5t^2}$ is strictly positive for every finite $t>0$, since $h>0$ throughout its own domain — so this particular $h(t)$ never has a stationary point at all, only an asymptote it approaches but never reaches in finite time. A candidate who reaches for "differentiate and set to zero" on this specific closed form is not making an arithmetic slip; they are applying a method that provably cannot terminate for this equation, which is precisely why the real mark scheme credits the algebraic long-run reading instead. Whether this exact extension — solve, then find a limiting value from the closed form — recurs on other WMA14 series is not yet knowable from the record this course has reviewed: it is confirmed on exactly one real question, and stated as exactly that below, not extrapolated into a general "always check for a limit" rule the record does not yet support.

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#5. Synthesis — every real exam angle this pair of spec points has actually been tested from

Per this course's own SIGNAL document, this bundled topic is confirmed seen in 4 of 7 series that received real question-level mining — stated exactly that way, as a count against a small, named denominator, never as a percentage or a predictive likelihood. No grade-boundary figure exists anywhere in this course's WMA14 source review for any topic, this one included, and none is stated here.

What this synthesis adds beyond the existing documents: the lesson file, method skeleton and frequency row each record that these questions exist and, in the lesson's case, walk two of them (one real, one original) through in full. This document is the first place in the technique-hub layer that sorts every real citation by which antiderivative family it belongs to (§2.1 vs §2.2) and separates the setup-stage failures (§3) from the solving-stage ones (§2) as two genuinely distinct skill clusters — the connective layer between the theory in §1 and the bare citation list the other documents already carry.

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#6. What is genuinely not evidenced — stated rather than invented

Three plausible-sounding extensions of this topic have zero citation anywhere in the facts bank's WMA14 review, and nothing above relies on them:

• A non-separable first-order equation, or any technique for one (integrating factors, for instance). Spec 6.4's own text scopes the whole spec-required technique to the separable case; no examiner report or mark scheme reviewed for WMA14 cites any other first-order solving method.
• A pure exponential-growth-or-decay model ($\frac{dN}{dt}=kN$), the textbook-canonical separable example. Despite being the most common illustration of this technique outside this course's own documents, it has no citation anywhere in this facts bank's WMA14 review — the two real anchors this document actually has (Jan 2026 Q3's rational-power model, and the tank-style questions behind Oct 2021 Q9) are both structurally different from it, and no claim above substitutes the familiar textbook example for a real one.
• A grade-boundary or likelihood figure for this topic specifically. None exists anywhere in this course's WMA14 source review, confirmed directly in the SIGNAL document's own §2 — restated here because a deep-dive is exactly the place a fabricated "worth N% of the paper" claim would otherwise creep in.

This is the honest shape of the evidence for this topic: real, and rich enough for a genuine two-family antiderivative taxonomy, a four-point setup-failure list, and one fully real closing extension — but built from five questions across four series, not a larger archive, and this document's length is scoped to that, not stretched past it.

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#Cross-references (cited, not duplicated)

• Full mechanism derivation of why separation is legal, both worked chains, both full marked solutions (the VERIDIAN-original tank problem and the real Jan 2026 shrub-height question), all six trap-taxonomy items and four MCQs: content/veridian/curriculum/wma14/separable-differential-equations.ts.
• The seven-line method skeleton this document expands into a citation-sorted taxonomy: codex-formula-and-method-bank.md §2.8.
• Per-question mark tariffs and the cross-topic "show that" integrity pattern (§2.5, §3 above): forge-exam-technique-guide.md.
• The "4 of 7 series" frequency figure (§5 above): signal-frequency-and-boundaries.md §1.
• The formula-booklet's absence of any solid-volume formula, used without re-derivation in §3: WMA14-verified-facts.md §3.
• The $\frac1a$ linear-inside scaling-factor slip, confirmed elsewhere on spec 6.3 and flagged here only as a transfer risk (§2.1): codex-formula-and-method-bank.md §6 (partial fractions).

#Sources used

research/veridian/WMA14-verified-facts.md §1 (spec 5.2 and 6.4 verbatim text), §3 (formula-booklet contents), §4 (mark-scheme conventions — M/A/B, ag, ft, cao), §5.5 (Jan 2024 Q4, the spec 5.1/5.2 boundary), §5.6 (this topic's five direct citations: Jan 2021 Q10, Oct 2021 Q2, Oct 2021 Q9, and the Jan 2026 Q3 addition with its full independent-verification note), §7 (topics-with-enough-material note), §8 (evidence-strength flags). The Jan 2026 Q3 mark-scheme fragments quoted in §2.1, §2.4 and §4 are drawn from the facts bank's own re-verified transcription of WMA14_01_2601_MS (Publications Code, Question Paper Log Number P87595A), independently re-derived with sympy per that document's own provenance note. No new primary Pearson document was fetched this session.