Differential equation

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

It has the general form:

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

Ordinary v. differential equation

An ordinary equation such as $x^2=9$ is solved by numbers; a differential equation such as $\dot{y}=2y$ is solved by functions $y(t)$.

$\dot{y}$ 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 $x(t)$,

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

is of order 2: the unknown $x$ appears together with its second derivative, while the net force $F$ is given.

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

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

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.

If the unknown function depends on several independent variables, the equation is a partial, as the function can only be partially differentiated according to a given variable.

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.

Solutions

Many differential equations have multiple solutions.

For example, $y=Ce^{2t}$ solves $\dot{y}=2y$ for every constant $C$, since

$$y=Ce^{2t} \ \Longrightarrow\ \dot{y}=2Ce^{2t}=2y$$

So the equation has infinitely many solutions, all in the family of $y=Ce^{2t}$. We call the whole solution family its general solution.

A general solution contains one or more arbitrary constants.

Prescribing values of the solution function and its derivatives at certain points fixes those constants and selects one particular solution from the general solution family.

For instance, for $\dot{y}=2y$, the single condition $y(0)=3$ fixes the constant:

$$\begin{gathered} 3=Ce^0 \\ C=3 \end{gathered}$$

If values are prescribed at a single point, this prescription is often called an initial condition; if they are prescribed at two different points, the prescriptions are often called boundary conditions.

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 describe complex relationships between quantities and their rates of change.

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 $u_{t}=\alpha u_{xx}$.

See also

• Ordinary differential equation
• Partial differential equation

Further reading

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