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 $\theta=\pi$

Linear equations and superposition

An equation is linear when its terms in the unknown are of the first degree; collecting them on the left gives

$$L[y]\equiv y^{(n)}+a_{n-1}(x)\,y^{(n-1)}+\cdots+a_{1}(x)\,y'+a_{0}(x)\,y=g(x)$$

where the coefficients $a_{0}$, …, $a_{n-1}$ depend on $x$ only. The operator $L$ is linear, $L[c_{1}y_{1}+c_{2}y_{2}]=c_{1}L[y_{1}]+c_{2}L[y_{2}]$, so solutions combine:

$$ \begin{cases} L[y_{1}]=L[y_{2}]=0 \ \Longrightarrow\ L[c_{1}y_{1}+c_{2}y_{2}]=c_{1}\cdot 0+c_{2}\cdot 0=0 & \text{(superposition)} \\ L[y_{h}]=0,\quad L[y_{p}]=g \ \Longrightarrow\ L[y_{h}+y_{p}]=0+g=g & \text{(homogeneous + particular)} \end{cases} $$

Hence the general solution of a linear equation is $y=y_{h}+y_{p}$: the homogeneous solution $y_{h}$ carries the $n$ arbitrary constants, and $y_{p}$ is any one solution of the full equation. Neither rule survives for a nonlinear equation, such as $y'=y^{2}$:

$$y_{1}'=y_{1}^{2},\quad y_{2}'=y_{2}^{2} \ \Longrightarrow\ (y_{1}+y_{2})'=y_{1}^{2}+y_{2}^{2}\neq (y_{1}+y_{2})^{2}$$

so the sum of two solutions is not a solution.

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