Differential equations

The idea

A separable differential equation $\frac{dy}{dx} = f(x)g(y)$ is solved by putting all the $y$ on one side and all the $x$ on the other: $\int \frac{1}{g(y)}\,dy = \int f(x)\,dx$ . Use the initial condition to find the constant.

Separate: $\frac{1}{g(y)}\,dy = f(x)\,dx$ .
Integrate both sides; add one constant.
Use the given values to find it, then make $y$ the subject.

Watch out Find the constant before you rearrange, and check the answer makes sense in context.

See it worked

Worked example

Solve $\frac{dy}{dx} = 2xy$ , given $y = 3$ when $x = 0$ .

$\int \frac{1}{y}\,dy = \int 2x\,dx$ , so $\ln y = x^2 + c$ .
$x = 0$ , $y = 3$ : $c = \ln 3$ .
$y = e^{x^2 + \ln 3} = 3e^{x^2}$ .

Answer $y = 3e^{x^2}$

Your turn

Question 1Non-calculator3 marks

Solve $\frac{dy}{dx} = 3y$ , given $y = 2$ when $x = 0$ .

Question 2Non-calculator3 marks

$\frac{dy}{dx} = \frac{x}{y}$ and $y = 2$ when $x = 0$ , with $y > 0$ . Find the exact value of $y$ when $x = 3$ .

Question 3Calculator allowed3 marks

A population satisfies $\frac{dN}{dt} = -0.2N$ , with $N = 500$ when $t = 0$ . Find $N$ when $t = 5$ , to the nearest whole number.

In past papers

38 official questions in 38 papers since 2018. The latest:

All A-level Maths past papers

A-level Maths Pro, Pearson Edexcel 9MA0

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