Mathematics · University
Second-order equations and resonance · revision notes
For constant coefficients, an exponential trial function reduces the homogeneous ODE to a polynomial.
Key ideas and conditions
Characteristic equation
For constant coefficients, an exponential trial function reduces the homogeneous ODE to a polynomial.
Initial data
A second-order equation usually requires two initial conditions for a unique local solution under standard hypotheses.
Repeated roots
A repeated root r needs both e^(rt) and te^(rt).
Resonance
A forcing term that duplicates a homogeneous solution requires an extra power of t in a particular-solution trial.
Explain without looking
- Explain why 2e^t does not add an independent solution direction to e^t.
- In an undamped forced oscillator at its natural frequency, what does the ideal resonant model permit?
Further reading and review references
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