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Nomenclature

The theory of differential equations is well developed and the methods used to study them vary significantly with the type of the equation.

Ordinary

An ordinary differential equation (ODE) is a differential equation in which the unknown function y (also known as the dependent variable) is a function of a single independent variable. In the simplest form, the unknown function is a real:

$y:\mathbb {R} \rightarrow \mathbb {R} \,,\quad y=y(x)$

and a differential equation of order n is in the form:

$F:\mathbb {R} ^{n+1}\rightarrow \mathbb {R} \,,\quad F\left(x,y,{\frac {dy}{dx}},{\frac {d^{2}y}{dx^{2}}}\cdots {\frac {d^{n}y}{dx^{n}}}\right)=0$

More generally, a set of unknown functions can be collected together in a vector-valued function:

$y:\mathbb {R} \rightarrow \mathbb {R} ^{m}\,,\quad \mathbf {y} =\mathbf {y} (x)=(y_{1}(x),y_{2}(x)\cdots y_{m}(x))$

and the differential equations of order n form an m-dimensional system:

$\mathbf {F} :\mathbb {R} ^{mn+1}\rightarrow \mathbb {R} \,,\quad \mathbf {F} \left(x,\mathbf {y} ,{\frac {d\mathbf {y} }{dx}},{\frac {d^{2}\mathbf {y} }{dx^{2}}}\cdots {\frac {d^{n}\mathbf {y} }{dx^{n}}}\right)={\boldsymbol {0}}$

that is

$\mathbf {F} =\left(F_{1}\left(x,\mathbf {y} ,{\frac {d\mathbf {y} }{dx}},{\frac {d^{2}\mathbf {y} }{dx^{2}}}\cdots {\frac {d^{n}\mathbf {y} }{dx^{n}}}\right),F_{2}\left(x,\mathbf {y} ,{\frac {d\mathbf {y} }{dx}},{\frac {d^{2}\mathbf {y} }{dx^{2}}}\cdots {\frac {d^{n}\mathbf {y} }{dx^{n}}}\right)\cdots F_{m}\left(x,\mathbf {y} ,{\frac {d\mathbf {y} }{dx}},{\frac {d^{2}\mathbf {y} }{dx^{2}}}\cdots {\frac {d^{n}\mathbf {y} }{dx^{n}}}\right)\right)=(0,0\cdots 0)$

Generalizing further, differential equations can be matrix-valued. An additional generalization is that the equation may be complex-valued; a complex differential equation, so the unknown function becomes:

$f:\mathbb {C} \rightarrow \mathbb {C} \,\quad f=f(z)$

in which the differential equations (and generalizations vectors and matrices etc.) take analagous forms to the above.

The most important cases for applications are first-order and second-order differential equations. For example, Bessel's differential equation

$x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+(x^{2}-\alpha ^{2})y=0$

(in which y is the dependent variable) is a second-order differential equation. In the classical literature also distinction is made between differential equations explicitly solved with respect to the highest derivative and differential equations in an implicit form.

Partial

A partial differential equation (PDE) is a differential equation in which the unknown function u is a multivariable function; a function of many independent variables, and the equation involves its partial derivatives. It's clumsy to write down the general form in ordinary notation, but multi-index notation is compact.

In the simplest form, the unknown function is a real function of r variables:

$u:\mathbb {R} ^{r}\rightarrow \mathbb {R} \,,\quad u=y(\mathbf {x} )=u(x_{1},x_{2}\cdots x_{r})\,\quad \mathbf {x} =(x_{1},x_{2}\cdots x_{r})$

and a differential equation of order n is in the form (using multi-index notation):

$E:\mathbb {R} ^{n+1}\rightarrow \mathbb {R} \,,\quad E\left(x^{\alpha },u,{\frac {\partial u}{\partial x^{\alpha }}},{\frac {\partial ^{2}u}{\partial x^{\beta }\partial x^{\alpha }}},\cdots {\frac {\partial ^{r}u}{\partial x_{r}\cdots \partial x_{2}\partial x_{1}}}\right)=0$

where

${\frac {\partial u}{\partial x^{\alpha }}}=\left({\frac {\partial u}{\partial x_{1}}},{\frac {\partial u}{\partial x_{2}}}\cdots {\frac {\partial u}{\partial x_{r}}}\right)$

The order is defined similarly to the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic equations, especially for second-order linear equations, is of utmost importance. Some partial differential equations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of mixed type.

Linear and non-linear

Both ordinary and partial differential equations are broadly classified as linear and nonlinear.

A differential equation is linear if the unknown function and its derivatives appear to the power 1 (products are not allowed) and nonlinear otherwise. The characteristic property of linear equations is that their solutions form an affine subspace of an appropriate function space, which results in much more developed theory of linear differential equations. Homogeneous linear differential equations are a further subclass for which the space of solutions is a linear subspace i.e. the sum of any set of solutions or multiples of solutions is also a solution. The coefficients of the unknown function and its derivatives in a linear differential equation are allowed to be (known) functions of the independent variable or variables; if these coefficients are constants then one speaks of a constant coefficient linear differential equation.

There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness).

Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).