Differential Equations · Lesson 09

Chapter practice

Bring the chapter together: choose between undetermined coefficients and variation of parameters, solve a forced problem with its conditions, classify a damped spring, compute a steady amplitude, and recognise resonance. No worked example sits above the answers.

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m y^{''} + c y^{'} + k y = F (t)

ω_{0} = \frac{k}{m}

y = y_{h} + y_{p}

F_{0} cos ω t

e^{- γ t}

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Intuition

Bring the chapter together: choose between undetermined coefficients and variation of parameters, solve a forced problem with its conditions, classify a damped spring, compute a steady amplitude, and recognise resonance. No worked example sits above the answers.

A sound engineer hearing a hum asks three things: what is pushing, at what frequency, and how much the room resists. Those are the three questions of this chapter.

Before you practise

Decide first what is asked. A particular solution comes from guessing the shape of the forcing, multiplied by $x$ when the guess solves the homogeneous equation, or from varying the constants; the conditions go on the whole solution, last. A spring $mx''+cx'+kx=F(t)$ is sorted by comparing $c^{2}$ with $4mk$ , and a cosine forcing leaves the steady amplitude below.

A(\omega)=\frac{F_{0}}{\sqrt{(k-m\omega^{2})^{2}+(c\omega)^{2}}}

Reminders

$y=y_{p}+C_{1}y_{1}+C_{2}y_{2}$ , with the constants found after $y_{p}$ is added.
Natural frequency $\omega_{0}=\sqrt{k/m}$ , period .
With $c>0$ every motion forgets its start and settles to the steady state at the driving frequency.
Without friction, driving at $\omega_{0}$ gives a response that grows like $t\sin\omega_{0}t$ .

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