Differential equation

A differential equation is an equation which relates an unknown function to its derivatives.

It has the general form:

$$F\!\left(x_1,\dots,x_m,\;u,\;\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_m},\;\frac{\partial^2u}{\partial x_1^2},\dots,\frac{\partial^\alpha u}{\partial x_1^{\alpha_1}\cdots\partial x_m^{\alpha_m}},\dots\right)=0, \qquad \alpha=(\alpha_1,\dots,\alpha_m)\in\mathbb N^m.$$

Regular v. differential equation

A regular equation such as $x^2=9$ is solved by numbers; a differential equation such as $\dot{y}=2y$ is solved by functions $y$.

$\dot{y}$ and $y'$ are alternative notations for $\frac{dy}{dx}$

Classification

Order

The order of the equation is the order $n$ of the highest derivative that appears.

For instance, Newton's second law of motion for a body of mass $m$ at position $r(t)$,

$$m\frac{d^{2}\vec r}{dt^{2}}=\vec F$$

is of order 2 with regard to $\vec r$.

Linearity

A differential equation is linear when all of the following conditions are satisfied:

The unknown function and its successive derivatives

• appear only to the first power,
• are never multiplied together,
• and are never the arguments of another function.

A linear differential equation of order $n$ has the general form

$$\sum_{|\alpha|\le m} a_\alpha(\mathbf{x})\,D^\alpha u(\mathbf{x})=f(\mathbf{x}), \qquad \mathbf{x}=(x_1,\dots,x_d).$$

Where:

$a_\alpha(\mathbf{x})$ and $f(\mathbf{x})$ are functions of $\mathbf{x}$;

$$ \alpha=(\alpha_1,\dots,\alpha_d), \qquad |\alpha|=\alpha_1+\cdots+\alpha_d; $$

and

$$ D^\alpha u= \frac{\partial^{|\alpha|}u} {\partial x_1^{\alpha_1}\cdots \partial x_d^{\alpha_d}}. $$

When $f(\mathbf{x})=0$, the differential equation is further said to be homogeneous.

Examples:

$$ \begin{cases} \dot{y}=2y & \text{linear} \\ \dot{y}=y^{2} & \text{nonlinear: square of the unknown function y} \\ \ddot{\theta}+\dfrac{g}{L}\sin\theta=0 & \text{nonlinear: sine of the unknown function} \theta(t) \end{cases} $$

Ordinary v. partial differential equation

A differential equation is ordinary if there is only one independent variable.

A differential equation is partial if there are several independent variables.

For example: The heat equation

$$\frac{\partial u}{\partial t}=\alpha\frac{\partial^{2}u}{\partial x^{2}}$$

for the temperature $u(x,t)$ of a one-dimensional metal bar is a partial differential equation.

Methods of solution

No general formula solves all differential equations, just as no general formula solves all algebraic equations.

Some simple differential equations can be solved algebraically:

• Ordinary differential equations:
  • First-order: separation of variables, integrating factor
  • linear equations with constant coefficients, of the second and higher orders: by their characteristic equation
• Partial differential equations: separation of variables
  • First-order: method of characteristics

Some methods exist to transform differential equations that cannot be directly solved into directly solvable forms:

• Power series
• Laplace transform

More complex differential equations that cannot be solved algebraically can be analysed otherwise:

• Numerical solution: Euler's method and its higher-order refinements step along the direction field, when only numbers are required.
• Qualitative analysis: equilibria, stability, and long-term behaviour, read directly from the equation.

Application

Differential equations can describe complex relationships between quantities and their rates of change in nature.

For instance,

• Newton's second law can be stated as $m\ddot{\mathbf{r}}=\mathbf{F}$
• The decay of a radioactive sample and the growth of an unchecked population both follow $\dot{y}=ky$
• Conduction of heat through a solid is described by the partial differential equation $u_{t}=\alpha u_{xx}$.

$\ddot{r}$, $u_{t}$ and $u_{xx}$ are alternative notations for $\frac{d^2r}{dt^2}$, $\frac{\partial u}{\partial t}$ and $\frac{\partial^{2}u}{\partial x^{2}}$.

See also

• Ordinary differential equation
• Partial differential equation

Further reading

• Differential Equation, Wolfram MathWorld
• 18.03SC Differential Equations, MIT OpenCourseWare