Topic summary · Further maths (FP2)

First-order Differential Equations summary

First-order differential equations are often solved by separating variables, then applying a condition to find the constant.

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Key learning points

Separate: $\int\frac{1}{g(y)}\,dy=\int f(x)\,dx$ when $\frac{dy}{dx}=f(x)g(y)$ .
Include $+C$ , then use the given condition.
Logarithms produce absolute values; rewrite as exponentials when useful.
Note any solutions lost by dividing through by an expression.

Worked example 1 — Two-step equation

Solve $3x+7=19$ .

Working

x=4

Undo +7, divide by 3.

Worked example 2 — Unknown on both sides

Solve $5x-2=3x+10$ .

Working

x=6

Collect like terms.

Worked example 3 — With brackets

Solve $2(x-3)=10$ .

Working

x=8

Expand or divide first.

Worked example 4 — Separate

Solve $\frac{dy}{dx}=3y$ for $y>0$ .

Working

y=Ae^{3x}

Separate, integrate, exponentiate; $A>0$ .

Scaffolded Learning Sheet and answers (PDFs)

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