1. Overview¶
We now introduce a new wrinkle into our ODEs: second derivatives. Together, first- and second-order ODEs comprise the overwhelming majority of differential equations used in mathematical models. To some degree this traces back to Newton’s laws of motion; in \(F=ma\), acceleration is the second derivative of position. In fact, \(F=ma\) is the original justification for the term “forcing function” that we have used for a while, as it plays the role of an actual external force acting on a system.
First-order ODEs are all about growth and decay. Second-order ODEs add oscillation to the party. They also require us to update some of our theoretical understanding of the structure of solutions.
1.1. Glossary¶
- amplitude–phase form¶
Expression of a sum of sin-cos or onjugate complex exponentials as an amplitude times a cosine with phase shift.
- angular frequency¶
Expression of frequency as radians per second, as opposed to cycles per second; mathematical default sense of frequency.
- beating¶
Slow modulation of amplitude, characteristic of an oscillator being forced at a frequency near its natural one.
- characteristic polynomial¶
For a second-order linear ODE with constant coefficients, a quadratic polynomial whose roots indicate two exponential functions that form a basis for solutions.
- characteristic values¶
For a second-order linear ODE with constant coefficients, the roots of the characteristic polynomial.
- constant-coefficient¶
Linear ODE in which the multipliers of the dependent variable and its derivatives are constants.
- critically damped¶
Free oscillator at the level of damping between underdamped and fully exponentially decaying solutions.
- damping coefficient¶
Nondimensional, nonnegative value indicating the relative amount of damping present in an oscillator. Denoted \(Z\).
- free oscillator¶
Naturally oscillating system not subjected to any external forces.
- gain¶
Ratio of the amplitudes of a particular solution and the harmonic forcing that produces it.
- natural frequency¶
Frequency at which an undamped free oscillator oscillates. Denoted \(\omega_0\).
- overdamped¶
Free oscillator with sufficiently large damping that solutions are purely exponentially decaying, i.e., having no oscillations.
- polar form¶
Expression of a complex number as its magnitude times a value on the unit circle, as expressed by Euler’s identity.
- pseudoresonance¶
Damped oscillator being forced at the frequency that produces maximal gain.
- resonance¶
Undamped oscillator being forced at its natural frequency, resulting in boundless linear growth in the amplitude of the solution.
- second-order IVP¶
Second order ODE plus initial conditions on the solution and its first derivative.
- second-order ODE¶
Ordinary differential equation that includes a second derivative (and no higher) of the unknown, dependent variable.
- simple harmonic motion¶
Pure sinusoidal motion of an undamped free oscillator.
- spring constant¶
Ratio of the restoring force of an ideal spring to the amount by which it is stretched from its neutral position.
- underdamped oscillator¶
Free oscillator with nonzero damping that forces the solution to zero, but not so large that it prevents all oscillation.