Playbook
Differential Equations
144 q - 3.35/paper - 38% HARD. Six subtopics that split cleanly by SOLUTION METHOD, which is exactly how the questions are set. Order/Degree/Formation (33 q, 24% HARD) is recognition work and near-free; Linear (Integrating Factor) at 63% HARD is where the chapter gets expensive.
- Questions in the bank
- 144
- q/paper (2024–25 shifts)
- 3.35
- Tagged HARD
- 38%
- Subtopics
- 6
Strand: Cornerstone
When you’ll see it
An equation containing a derivative — either to be classified (order, degree, formation) or to be solved, plus the growth, decay and cooling word problems that reduce to one.
How this chapter is tested
144 q at 3.35 per paper and 38% HARD. The structural fact that should shape your preparation is that the six subtopics split by SOLUTION METHOD — variable-separable, homogeneous, linear with an integrating factor — and that is exactly how the exam sets them. So the first move on any stem is not to solve, it is to CLASSIFY. Get the classification right and the solution is a standard procedure; get it wrong and you lose the whole 1.8 minutes.
Order, Degree, Formation of ODE, and Verification of Solutions is 33 q at 24% HARD and is nearly free. It asks for recognition, not integration: read the highest derivative, read its power, count the arbitrary constants. Take this block first — it is a quarter of the chapter's questions at two thirds of the chapter's difficulty rate.
Linear Differential Equations (Integrating Factor) is where the chapter gets expensive: 24 q at 63% HARD, the highest rate here. Newton's Law of Cooling is only 5 q but runs 60% HARD, so it is genuinely optional. The three mid-weight blocks — Growth and Decay (33 q, 27%), Variable-Separable (33 q, 39%) and Homogeneous and Reducible (16 q, 38%) — are where the reliable marks are.
Verification questions are the clearest place in the subject where no negative marking changes tactics. If the stem gives a differential equation and four candidate solutions, differentiating each candidate and substituting is mechanical and always terminates. Solving the equation from scratch may not.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Order, degree, formation, and verification
Order is the highest derivative present; degree is its power once the equation is made polynomial in derivatives. Forming an ODE from a family means differentiating as many times as there are arbitrary constants. 33 q at 24% HARD.
Variable-separable equations
Get all the y with dy and all the x with dx, then integrate both sides and keep one arbitrary constant. 33 q at 39% HARD. The integration, not the separation, is usually the hard part — which is why Indefinite Integration comes first.
Homogeneous and reducible equations
Recognise that every term has the same total degree, substitute y = vx, and the equation becomes separable. Reducible forms are a shift of origin away from homogeneous. 16 q at 38% HARD.
Linear equations and the integrating factor
Put the equation in the standard first-order linear form, build the integrating factor, multiply through, and integrate. 24 q at 63% HARD — the hardest block in the chapter and the one to learn once the easier three are secure.
Growth, decay, and continuous models
Word problems whose rate is proportional to the amount present. They all reduce to the same separable equation; the work is translating the sentence and fixing the constant from the given condition. 33 q at 27% HARD.
Newton's law of cooling
A named special case of the same proportional-rate model, with the ambient temperature as an offset. 5 q at 60% HARD — low volume, high cost, so it is a reasonable thing to leave until everything else is done.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.
Degree quoted for an equation that has none
Degree is only defined once the equation is polynomial in its derivatives. If a derivative sits inside a radical, a trigonometric function or an exponential and cannot be cleared, degree is not defined — and a numeric option is offered anyway.
Wrong count of arbitrary constants
The order of the ODE formed from a family equals the number of independent arbitrary constants, not the number of letters in the equation. A fixed parameter dressed up as a constant inflates the order by one.
Integrating factor built for the wrong variable
Some stems are linear in x as a function of y, not y as a function of x. Applying the standard form to the wrong variable produces a clean-looking answer that solves a different equation.
Constant of integration dropped or misplaced
Options that differ only in where the arbitrary constant sits, or in whether it is inside a logarithm. Apply the given initial condition and the ambiguity disappears.
Learn it before you drill it
This chapter has full teaching notes — foundations, worked examples, self-checks and a per-subtopic mastery checkpoint. Read the notes once, then drill subtopic by subtopic below.
Differential Equations notesDrill every differential equations question
144 questions from the bank, scoped to 6 bundled subtopics.
Drill one subtopic at a time
The 6 subtopics this playbook covers, in catalog order.
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