Ordinary differential equation

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

General form

Solved for the highest derivative, an ordinary differential equation of order $n$ reads

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

and, written implicitly, $F\bigl(x,y,y',y'',\ldots,y^{(n)}\bigr)=0$. The order $n$ is the order of the highest derivative that appears; the equation of the first order $n=1$ is treated in 1st order ODE.

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

Labelling the successive derivatives of the unknown as new unknowns turns an equation of order $n$ into a system of $n$ equations of order 1:

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

The pendulum illustrates the reduction. Its angle $\theta(t)$ obeys the equation of motion

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

which involves the second derivative; taking the angular velocity $v=\dot{\theta}$ as the second unknown gives the equivalent first-order system

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

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

No information is lost in the reduction, so the geometric, numerical, and qualitative methods available for first-order equations apply at every order (Euler's method, Qualitative methods).

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.

Solutions and initial conditions

A solution is a function that satisfies the equation identically, as $\cos\omega_{0}t$ satisfies the harmonic oscillator equation:

$$x=\cos\omega_{0}t \ \Longrightarrow\ \ddot{x}=-\omega_{0}^{2}\cos\omega_{0}t=-\omega_{0}^{2}x \ \Longrightarrow\ \ddot{x}+\omega_{0}^{2}x=0$$

Together with $\sin\omega_{0}t$ it gives the general solution of that equation, with one arbitrary constant for each order:

$$x(t)=C_{1}\cos\omega_{0}t+C_{2}\sin\omega_{0}t$$

Fixing the constants requires as many conditions as the order. For a mass pulled to the position $x_{0}$ and released from rest at $t=0$,

$$x(0)=C_{1}=x_{0} \ \Longrightarrow\ \dot{x}(t)=-C_{1}\omega_{0}\sin\omega_{0}t+C_{2}\omega_{0}\cos\omega_{0}t \ \Longrightarrow\ \dot{x}(0)=C_{2}\omega_{0}=0$$

so $C_{2}=0$, and the particular solution is the oscillation

$$x(t)=x_{0}\cos\omega_{0}t$$

Imposed at a single point, the $n$ conditions

$$y(x_{0})=y_{0},\qquad y'(x_{0})=y_{1},\qquad \ldots,\qquad y^{(n-1)}(x_{0})=y_{n-1}$$

form an initial value problem; at the first order the single condition is

Conditions imposed at two different points are boundary conditions, and the problem is a boundary value problem, which may have no solution or infinitely many.

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