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Differential Equations Types

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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)

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)

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)

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

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}

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.

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}

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.

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.

Autonomous Differential Equation

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

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

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.

Non-Linear Differential Equation

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

y'' + yy' = 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)

Partial Differential Equation

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

u_{t} = k u_{xx}

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

Bessel's Differential Equation

An equation of the form:

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

where

n

is a constant.

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)

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

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