Differential Equations Applications

Simple harmonic motion

Some differential equations have solutions with sines and cosines, which describe oscillating behaviour. Simple harmonic motion is described by a purely sinusoidal solution, and can be applied to many physical situations.

The differential equation for simple harmonic motion is:

Where represents the displacement from the equilibrium position (where the acceleration is zero). Solving this differential equation involves solving the auxillary equation , giving a general solution of:

The general solution can also be written as:

Where is the amplitude, and is the phase shift.

For certain cases, the solution is simplified as either the or constants is clearly zero. For example:

If an object is initially at rest at maximum displacement from the equilibrium position, then the general solution is .
If an object is initially at the equilibrium position, then the general solution is .

There are some useful definitions for simple harmonic motion:

The equilibrium position is the position around which the object oscillates.
The amplitude is the maximum distance from the equilibrium position.
The period is the time after which the motion repeats.
The angular frequency is (omega).

The period and angular frequency are related, as the motion repeats when :

The object has maximum speed as it goes through the equilibrium position. The object is instantaneously at rest when at maximum displacement from the equilibrium position.

For a object with simple harmonic motion with amplitude , velocity and displacement are related by:

Proof of the above result:

Let (we can adjust when we start so this is true).
Differentiating, we get .
Rearranging, we get .
Substituting in , we get .
Giving as required.

Damping

Objects often move against a resistive force, e.g. air resistance or water resistance. One way to account for these forces is to model them as proportional to the speed, acting against the motion:

Where is a constant.

By applying :

This is called damped simple harmonic motion. The solutions to this differential equation depend on the discriminant of the auxiliary equation, from Differential Equations.

The cases are:

describes overdamping, where there are no oscillations. The general solution is , where and are roots of the auxiliary equation.
describes critical damping. The general solution is , as if the discriminant is zero, the auxiliary equation will have a repeated root at .
describes underdamping, where there is oscillatory behaviour. The general solution is , where is the imaginary part of the roots to the auxiliary equation. The roots are given by , so the real part is and the imaginary part is .

Linear systems

Systems of coupled differential equations can describe model situations such as predator-prey models, where the rate of change of each independent variable is related to the other variable. For linear systems with first-order equations, these can be solved by eliminating variables to form a second-order differential equation in one variable and solving using standard methods.