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Playbook

Differential Equations

Three solving methods and the pages that disguise them. The answer is often a second step: a value, a maximum or an integral of the solution.

Questions in the bank
180
q/paper in 2025–26
1.14
Numeric answer
26%
Notes pages
9

Tier: Core

When you’ll see it

A relation between y, x and dy/dx, a curve fixed by its tangent or normal, a growth or cooling rate, or a function defined by an integral of itself.

How this chapter is tested

The chapter comes down to three solving methods — separate the variables, put y = vx, or use the integrating factor — plus the pages that disguise one of them. Separable and linear equations are routine: the work is the integral and the constant from the given point.

Time goes on recognising the form. An equation linear in x rather than y, a Bernoulli equation, a substitution for tan y or e^(sin y), or an integrating factor that is just a denominator: each becomes routine after one move. Questions that never print an equation — an integral equation or a limit — produce one when differentiated.

The answer is often a second step: a value of the solution, its maximum, or an integral of it, which draws on application of derivatives and definite integration. Forming an equation and reading its order and degree is short, but the degree is read only after radicals are cleared.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Forming an equation

    Differentiate once per arbitrary constant and eliminate them; read the degree only after clearing radicals.

  • Separating the variables

    Put every y on one side, integrate, fix C; when the slope depends on ax + by + c, substitute t = ax + by + c.

  • Homogeneous equations

    y = vx turns dy/dx = f(y/x) into a separable equation; shift the origin first when the two lines meet.

  • Linear equations

    For dy/dx + Py = Q, multiply by e^(∫P dx) so y · IF = ∫ Q · IF dx; a long P is often the derivative of a log.

  • Reducible to linear

    Take x as the unknown function of y, divide a Bernoulli equation by a power of y, or substitute for tan y or e^(sin y).

  • Using the solution

    Fix the constant by a limit where no point is given, then maximise, differentiate or integrate the solution.

  • Hidden equations and rates

    Differentiate an integral equation and get the starting value from the lower limit; translate tangent, normal, growth and cooling conditions.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • Standard form first

    In xy′ + 2y = x², P is 2/x, not 2. The coefficient of y′ must be 1 before the integrating factor is found.

  • The tan x integrating factor

    ∫ tan x dx = ln sec x, so P = tan x gives the factor sec x and P = −tan x gives cos x. Swapping them is the common slip.

  • The constant added too late

    From ln y = x² + C the solution is y = Ae^(x²), not e^(x²) + C.

  • Cooling the whole temperature

    In Newton's law of cooling the excess T − A decays like e^(−kt), not T itself.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Differential Equations notes

Drill every Differential Equations question

180 questions from the bank, across 9 subtopics.

Drill the 180 questions

Drill one subtopic at a time

The 9 subtopics, in teaching order.

Related playbooks

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