The idea

Differential equations

Solving first-order separable differential equations by separating the variables and integrating both sides, finding a particular solution from a boundary condition, and modelling rates of change.

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Pure · Integration

Differential equations

Solving first-order separable differential equations by separating the variables and integrating both sides, finding a particular solution from a boundary condition, and modelling rates of change.

Why it works

An equation about a rate

A differential equation is an equation involving a derivative — it tells you the rate at which something changes rather than its value directly. Solving it means recovering the original relationship

y

from

\dfrac{dy}{dx}

, which is exactly what integration does.

Separate the variables

The type you can solve at A-level is separable: the right-hand side factorises into a part in

x

and a part in

y

\frac{dy}{dx} = f(x)\,g(y)

. Get all the

y

's (with

dy

) on one side and all the

x

's (with

dx

) on the other, then integrate both sides:

\int \frac{1}{g(y)}\,dy = \int f(x)\,dx

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