Ordinary differential equation

An ordinary differential equation (ODE) is a differential equation whose unknown function depends on a single independent variable.

General form

An ordinary differential equation of order $n$ has the general form

$$F\bigl(x,y,y',y'',\ldots,y^{(n)}\bigr)=0$$

Example:

A mass $m$ attached to a spring of stiffness $k$, displaced from its rest position and released, obeys the second-order equation

$$\frac{d^{2}x}{dt^{2}}+\omega_{0}^{2}x=0$$

where $\omega_{0}=\sqrt{k/m}$ is the angular frequency of the oscillation.

Reduction to a first-order system

For an ordinary differential equation, considering successive derivatives of the unknown function as new unknown functions may reduce the order of the original equation without losing any information:

$$F\bigl(x,y,y',y'',\ldots,y^{(n)}\bigr)=0 \Longrightarrow \begin{cases} y_2=y_1' \\ y_3=y_2' \\ \vdots \\ y_n=y_{n-1}' \\ y_n'=f(x,y_1,y_2,\ldots,y_n) \end{cases}$$

Example:

A pendulum obeys the equation of motion

$$\frac{d^{2}\theta}{dt^{2}}+\frac{g}{L}\sin\theta=0$$

which involves the second derivative.

To reduce the equation to a system of two first degree equations, introduce the angular acceleration $\omega=\frac{d\theta}{dt}$ as a second unknown:

$$\begin{cases} \omega=\frac{d\theta}{dt} \\ \frac{d\omega}{dt}=-\frac{g}{L}\sin\theta \end{cases}$$

From equation 2 it can be inferred that equilibria is reached at $\theta=0$ or $\theta=\pi$

See also

• 1st order ODE
• 2nd order ODE
• Partial differential equation
• Euler's method
• Qualitative methods
• Laplace transform
• Power series

Further reading

• Ordinary differential equation, Wikipedia
• Ordinary Differential Equation, Wolfram MathWorld
• 18.03SC Differential Equations, MIT OpenCourseWare