Practice question · Sort into groups
Sort each solution behaviour by the roots of its characteristic equation.
Groups: Two real negative roots · Complex roots, negative real part · Pure imaginary roots
- Decaying oscillation
- Smooth decay, no oscillation
- A struck tuning fork ringing down
- Steady oscillation, constant amplitude
Hints
- Real part → growth or decay. Imaginary part → oscillation. Combine the two readings.
- A tuning fork rings (imaginary part present) and fades (negative real part).
Show the answer
Two real negative roots: Smooth decay, no oscillation
Complex roots, negative real part: Decaying oscillation, A struck tuning fork ringing down
Pure imaginary roots: Steady oscillation, constant amplitude
Why
The complex plane is a behaviour map: left half = decay, right half = growth, off-axis = oscillation. Engineers judge the stability of aircraft and circuits by plotting the roots and checking which half-plane they occupy.
Practise Second-Order Linear ODEs
The app has 9 more questions on this lesson, and keeps your place in the course. Physics I is free to start.
More questions on Second-Order Linear ODEs
- For y'' + by' + ky = 0 (a damped oscillator), the roots of the characteristic equation are complex with…
- The damped oscillator’s behaviour changes qualitatively as damping increases, passing through a critical…
- Order the steps for solving a second-order linear ODE with constant coefficients.
- The general solution of a second-order ODE has two arbitrary constants, and a first-order one has a single…