Introduction
A first-order differential equation (ODE) relates an unknown function to its derivative . In many 9709 Paper 3 questions, such an equation arises either directly or from a modelling context (linking back to Forming Differential Equations), and you are then asked to solve it.
The most tractable class of first-order ODEs for A-Level purposes is the separable type, where the variables and can be algebraically arranged onto opposite sides of the equation before integrating. Mastery of this technique is essential: it regularly appears as a multi-part question worth 7–10 marks, and every integration technique from topic 3.5 (partial fractions, substitution, trigonometric identities, etc.) may be called upon within a single question.
Core Concept
A first-order ODE is separable if it can be written in the form
The method consists of four steps:
Step 1 — Separate. Rearrange so that all -terms (including ) are on the left and all -terms (including ) are on the right:
Step 2 — Integrate both sides simultaneously, introducing a single arbitrary constant :
Step 3 — Simplify. Express the relationship between and as clearly as possible (sometimes an explicit formula , sometimes an implicit relation).
Step 4 — Apply an initial condition (if given) to evaluate , yielding the particular solution.
The result before applying a condition is called the general solution — it represents a family of solution curves parameterised by .
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