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Differential Equations Types

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Coupled Differential Equations

A system where two or more differential equations are linked together through their dependent variables. Example:

\begin{cases} \frac{dx}{dt} = f(x, y) \\ \frac{dy}{dt} = g(x, y) \end{cases}

Bessel's Differential Equation

An equation of the form:

x^2y'' + xy' + (x^2 - n^2)y = 0

where

n

is a constant.

Non-Linear Differential Equation

An equation that involves a nonlinear combination of functions and their derivatives. Example:

y'' + yy' = 0

Inhomogeneous Differential Equation

A linear differential equation that contains a term that is not a function of the dependent variable and its derivatives alone. Example:

y'' + p(x)y' + q(x)y = g(x)

where

g(x) \neq 0

Bernoulli Differential Equation

A nonlinear differential equation of the form:

\frac{dy}{dx} + p(x)y = q(x)y^n

where

n \neq 0, 1

Partial Differential Equation

An equation involving partial derivatives of a function of multiple independent variables. Example:

u_{t} = k u_{xx}

Fourier's Heat Equation

A type of second-order linear partial differential equation of the form:

\frac{\partial u}{\partial t} - \alpha \frac{\partial^2 u}{\partial x^2} = 0

where

\alpha

is thermal diffusivity.

Second-Order Linear Differential Equation

A second-order differential equation that is linear in the unknown function and its derivatives. Example:

y'' + p(x)y' + q(x)y = r(x)

Legendre Differential Equation

An equation of the form:

(1 - x^2)y'' - 2xy' + n(n+1)y = 0

where

n

is a non-negative integer.

Laplace's Differential Equation

A second-order linear partial differential equation of the form:

\nabla^2 f = 0

where

\nabla^2

is the Laplacian.

Homogeneous Differential Equation

A differential equation where every term is a homogeneous function of the same degree of the dependent variable and its derivatives. Example:

x\frac{dy}{dx} = f(y/x)

First-Order Linear Differential Equation

A first-order differential equation with a degree of one in both the function and its derivative. Example:

\frac{dy}{dx} + P(x)y = Q(x)

Linear Differential Equation

An equation involving derivatives of a function which does not contain products or powers of the function or its derivatives. Example:

\frac{dy}{dx} + P(x)y = Q(x)

Autonomous Differential Equation

A differential equation that does not explicitly depend on the independent variable. Example:

\frac{dy}{dx} = f(y)

Riccati Differential Equation

A nonlinear differential equation of the form:

y' = q_0(x) + q_1(x)y + q_2(x)y^2

Wave Equation

A second-order linear partial differential equation of the form:

\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}

where

c

represents the wave speed.

Hypergeometric Differential Equation

A second-order linear differential equation of the form:

x(1 - x)y'' + [c - (a + b + 1)x]y' - aby = 0

where

a, b, c

are constants.

Ordinary Differential Equation

A differential equation containing one independent variable and its derivatives. Example:

\frac{d^2y}{dx^2} + p(x)\frac{dy}{dx} + q(x)y = 0

Separable Differential Equation

A differential equation in which the variables can be separated so that all terms involving one variable are on one side of the equation and all terms involving the other variable are on the other side. Example:

\frac{dy}{dx} = g(x)h(y)

Exact Differential Equation

An equation where there exists a function of two variables whose partial derivatives generate the terms in the equation. Example:

M(x, y)dx + N(x, y)dy = 0

where

\frac{∂M}{∂y} = \frac{∂N}{∂x}

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