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Topic 8 of 8 · Form 6

Second-order differential equations

Auxiliary equation am² + bm + c = 0:

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Homogeneous equations: ay″ + by′ + cy = 0

Auxiliary equation am² + bm + c = 0:

  • Distinct real roots m₁, m₂: y = Ae^(m₁x) + Be^(m₂x).
  • Repeated root m: y = (A + Bx)e^(mx).
  • Complex roots p ± qi: y = e^(px)(A cos qx + B sin qx).

Worked example: Solve y″ − 5y′ + 6y = 0.

m² − 5m + 6 = 0 → m = 2, 3 → y = Ae^(2x) + Be^(3x)

Non-homogeneous equations: ay″ + by′ + cy = f(x)

General solution = complementary function + particular integral.

  • f(x) a constant or polynomial → try a polynomial of the same degree.
  • f(x) = ke^(px) → try λe^(px) (multiply by x if p is a root of the auxiliary equation).
  • f(x) = k cos px or k sin px → try λ cos px + μ sin px.

Worked example: Solve y″ − 5y′ + 6y = 12 with y = 3 and y′ = 1 when x = 0.

PI: y = 2 (since 6y = 12). General: y = Ae^(2x) + Be^(3x) + 2

y(0) = 3: A + B = 1; y′(0) = 1: 2A + 3B = 1 → B = −1, A = 2

y = 2e^(2x) − e^(3x) + 2

Applications

Simple harmonic motion (y″ + ω²y = 0 gives y = A cos ωt + B sin ωt), damped oscillations, and electrical circuits.

Substitutions

Some equations reduce to standard forms using a given substitution (e.g. x = eᵗ for Euler-type equations).

Exam tip: Find the complementary function first – it tells you whether the usual trial particular integral needs adjusting.

Check yourself

  1. Solve y″ + 9y = 0.
  2. Solve y″ − 4y′ + 4y = 0.
  3. What trial function would you use for a particular integral of y″ + y = 3x?
Show answers
  1. y = A cos 3x + B sin 3x
  2. y = (A + Bx)e^(2x)
  3. y = ax + b (gives y = 3x)
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