A-Level — mathsAdvanced1 min read

Differential Equations

Solve first-order differential equations at A-Level using separation of variables.

Tutor AI Team2026-02-25T01:57:10.944Z

Differential equations relate a function to its derivatives. A-Level focuses on first-order separable equations solved by separation of variables.

Separation of Variables

$\frac{dy}{dx} = f(x)g(y)$ → $\frac{1}{g(y)}\,dy = f(x)\,dx$ → Integrate both sides.

Problem

$\frac{dy}{dx} = 3x^2 y$ .

$\frac{1}{y}\,dy = 3x^2\,dx$ → $\ln|y| = x^3 + C$ → $y = Ae^{x^3}$ .

Solution

Problem

$\frac{dy}{dx} = \frac{x}{y}$ , $y(0) = 2$ .

$y\,dy = x\,dx$ → $\frac{y^2}{2} = \frac{x^2}{2} + C$ .

$y(0) = 2$ : $2 = C$ . So $y^2 = x^2 + 4$ → $y = \sqrt{x^2 + 4}$ .

Solution

Problem

$\frac{dP}{dt} = kP$ (exponential growth). Solution: $P = P_0 e^{kt}$ .

Solution

1. Solve $\frac{dy}{dx} = xy^2$ .
1. Solve $\frac{dy}{dx} = \frac{e^x}{y}$ , $y(0) = 1$ .
1. A population satisfies $\frac{dN}{dt} = 0.05N$ . $N(0) = 1000$ . Find $N(t)$ .

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Tags:

#a-level#maths#pure#calculus#differential-equations

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