Second-Order Homogeneous Equations

Key idea

Use the characteristic equation.

In depth

For ay′′ + by′ + cy = 0, try y = e^(rt): ar² + br + c = 0. Two real roots give C₁e^(r₁t) + C₂e^(r₂t); a repeated root gives (C₁ + C₂t)e^(rt); complex roots α ± βi give e^(αt)(C₁cos βt + C₂sin βt).

Worked example

y′′ + 4y = 0: r = ±2i, so y = C₁cos 2t + C₂sin 2t.

Formula

Real-world uses

• Modelling damped springs in car suspensions and LC circuits in electronics

Common mistake

Forgetting the t factor for repeated roots.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Differential Equations, Abstract Algebra & Applied Math

• Exact Equations – M dx + N dy = 0 with Mᵧ = Nₓ.
• Bernoulli Equations – Substitute v = y¹⁻ⁿ to linearise.
• Runge–Kutta Methods – Higher-accuracy numerical solvers.
• Undetermined Coefficients – Guess the particular solution form.
• Variation of Parameters – General method for particular solutions.
• Mechanical Vibrations – Springs, damping and resonance.