Topic summary · Further maths (FP2)

Second-order Differential Equations summary

Second-order linear DEs with constant coefficients: auxiliary equation, complementary function, and particular integral.

Open Learning Sheet — Second-order Differential Equations Download Second-order Differential Equations answers PDF

Files: FP2-Second-order Differential Equations.pdf · FP2-Second-order Differential Equations answers.pdf · free public downloads · diagrams in the PDFs

Key learning points

Solve $am^2+bm+c=0$ for the complementary function.
Choose a particular integral matching the RHS (adjust if it overlaps the CF).
General solution = CF + PI; apply conditions for constants.
Repeated/complex roots change the CF structure.

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 — Auxiliary

For $y''-3y'+2y=0$ , solve the auxiliary equation.

Working

m=1\text{ or }2

CF $=Ae^{x}+Be^{2x}$ .

Scaffolded Learning Sheet and answers (PDFs)

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Open Learning Sheet Answers PDF

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