First Order Differential Equations Topic Review summary

Key learning points

• Separable: $\\frac{dy}{dx}=g(x)h(y)\\Rightarrow\\int\\frac{1}{h}\\,dy=\\int g\\,dx$.
• Linear first-order: $\\frac{dy}{dx}+P(x)y=Q(x)$ with integrating factor $e^{\\int P\\,dx}$.
• Find the general solution $+C$, then use a condition for the particular solution.
• Check by differentiating your answer.
• Homogeneous substitutions appear in some FP courses.

Worked example 1 — Separable

$\\frac{dy}{dx}=xy$.

Working

Separate and integrate.

Worked example 2 — IF

Integrating factor for $y'+2y=e^x$.

Working

Multiply through, recognise product rule.

Worked example 3 — Practice point 1

Separable: $\\\\frac{dy}{dx}=g(x)h(y)\\\\Rightarrow\\\\int\\\\frac{1}{h}\\\\,dy=\\\\int g\\\\,dx$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Practice point 2

Linear first-order: $\\\\frac{dy}{dx}+P(x)y=Q(x)$ with integrating factor $e^{\\\\int P\\\\,dx}$.

Working

Attempt without notes, then check the answers PDF.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.