MTH2032 Chap.4 Free Oscillations and Particular Solutions
Free Oscillations and Particular Solutions
Week 4 applies the constant-coefficient theory to a mass on a spring and then moves to non-homogeneous equations. Free oscillations come in three damping regimes decided by comparing c² with 4km.
Particular solutions are found by undetermined coefficients, guided by a table of trial functions and a modification rule for terms that already solve the homogeneous equation, or by variation of parameter, which works for any right-hand side once a basis is known. A unit check closes every model, since the constants k and c carry physical dimensions that must balance.
What this chapter covers
- 01
Hooke's law and my″ + ky = 0
- 02
Natural frequency and period
- 03
Under, critical and over damping
- 04
Dimensional analysis of k and c
- 05
Trial functions and the modification rule
- 06
Variation of parameter for second order
Worked example · free
Period and motion of an undamped spring
- 1The model is 0.5y″ + 8y = 0, so y″ + 16y = 0.
- 1ω0 = √(k/m) = 4 rad/s, so y = c1 cos 4t + c2 sin 4t.
- 1y(0) = 0.05 gives c1 = 0.05, and y′(0) = 0 gives c2 = 0.
- 1So y = 0.05 cos 4t, with period 2π/4 ≈ 1.571 s and frequency 4/(2π) ≈ 0.637 Hz.
Key terms
- Natural Frequency
- The frequency ω0/(2π) at which an undamped mass-spring system oscillates when left alone.
- Critical Damping
- The case c² = 4km, in which the mass returns to equilibrium fastest without oscillating.
- Trial Solution
- An assumed form for a particular solution whose unknown coefficients are found by substitution.
- Variation of Parameter
- A method that replaces the constants of a homogeneous solution by functions to produce a particular solution.
Free Oscillations and Particular Solutions FAQ
How do I decide which damping case I am in?
Compare c² with 4km from the equation my″ + cy′ + ky = 0. If c² is smaller the roots are complex and the motion oscillates in a decaying envelope; if they are equal there is a repeated root and critical damping; if c² is larger the roots are real and the motion is over-damped, written with cosh and sinh.
When does the modification rule apply?
Whenever a term of the initial trial already solves the homogeneous equation. Multiply the trial by x to the power of the multiplicity of the corresponding characteristic root. Forcing at the natural frequency of an undamped spring is the standard example, and it is why resonance produces a factor t in the solution.
When should I use variation of parameter instead of undetermined coefficients?
Use it when the coefficients are not constant or the right-hand side is not in the trial table, such as a secant or cosecant. It needs a known basis and the equation in standard form. If a question names the method, use that one and say so, because the problem-set solutions give marks for stating the method used.
How do I check a particular solution quickly?
Differentiate the trial twice and substitute it into the left-hand side of the equation. Collect terms by function type, such as all exponentials together, and compare with the right-hand side. If anything is left over, the trial or a coefficient is wrong. The check takes a minute and catches most sign and factor errors.
Exam move
Split revision into models and methods. For models, rebuild the mass-spring equation from Newton's second law and Hooke's law once, then practise reading m, c and k from a word problem with units, classifying the damping and fitting initial data. Add a unit check to every answer: k in kilograms per second squared, c in kilograms per second, and the product ω0t unitless.
For methods, memorise the trial table as patterns: exponential gives exponential, polynomial gives polynomial of the same degree, sine or cosine gives both. Then drill the modification rule on cases where the forcing exponent or frequency is a characteristic root, and verify each particular solution by substitution.
For variation of parameter, practise the full pipeline: standard form, basis, Wronskian, the two-by-two system, integrals, assembly. Choose one example with a non-constant coefficient so that you also practise dividing through by the leading coefficient before reading off the right-hand side.
When both methods are possible, use undetermined coefficients for speed and keep variation of parameter for equations with variable coefficients or unusual forcing terms.
Working through Free Oscillations and Particular Solutions in MTH2032? Sia is AskSia’s AI Mathematics tutor — ask any MTH2032 Free Oscillations and Particular Solutions question and get a clear, step-by-step explanation grounded in how MTH2032 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
