Second Order Differential Equations Topic Review summary

Key learning points

• Auxiliary equation for $ay''+by'+cy=0$; roots → complementary function.
• Repeated roots → $(A+Bx)e^{mx}$; complex roots → $e^{\\alpha x}(A\\cos\\beta x+B\\sin\\beta x)$.
• Particular integral: try forms matching the RHS (adjust if it overlaps the CF).
• General solution $=\\text{CF}+\\text{PI}$; apply conditions for unknowns.
• Verify by substitution when time allows.

Worked example 1 — Auxiliary

$y''-3y'+2y=0$.

Working

Distinct real roots.

Worked example 2 — PI

RHS is $e^{3x}$ and $3$ not a root — try.

Working

Substitute to find $C$.

Worked example 3 — Practice point 1

Auxiliary equation for $ay''+by'+cy=0$; roots → complementary function.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Practice point 2

Repeated roots → $(A+Bx)e^{mx}$; complex roots → $e^{\\\\alpha x}(A\\\\cos\\\\beta x+B\\\\sin\\\\beta x)$.

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.