Differential Equations

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Table of contents
1 Ordinary differential equation
2 Existence and uniqueness of solution
3 Examples of ODE
4 Homogeneous linear ODEs with constant coefficients
5 Partial differential equation
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Ordinary differential equation

In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable.

A simple example is Newton's second law of motion, which leads to the differential equation

m \frac{d^2 x}{dt^2} = f(x)\,,

for the motion of a particle of mass m. In general, the force f depends upon the position of the particle x, and thus the unknown variable x appears on both sides of the differential equation, as is indicated in the notation f(x).

Ordinary differential equations are to be distinguished from partial differential equations where there are several independent variables involving partial derivatives. Many famous mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoullis, Riccati, Clairaut, d'Alembert and Euler.

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Existence and uniqueness of solution

A solution of a differential equation is a function y(x) whose derivatives satisfy the equation. Such a function is not guarranted to exist and, if it does exist, it is usually not unique.

The existence and uniqueness of solutions to ordinary differential equations can be demonstratd using the Picard-Lindelof theorem.

There is a general theorem (the Cauchy-Kovalevskaya theorem) that states that the cauchy problem for any partial differential equation that is analytic in the unknown function and its derivatives has a unique analytic solution.sfsfsf

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Examples of ODE

Much study has been devoted to the solution of ordinary differential equations. In the case where the equation is linear, it can be solved by analytical methods. Unfortunately, most of the interesting differential equations are non-linear and, with a few exceptions, cannot be solved exactly. Approximate solutions are arrived at using computer approximations (see numerical ordinary differential equations).


Let y represent an unknown function of x, and let 

  y', y'',\ \dots,\ y^{(n)}

denote the derivatives 

   \frac{dy}{dx},\ \frac{d^{2}y}{dx^2},\ \dots,\ \frac{d^{n}y}{dx^{n}}.

' An ordinary differential equation (ODE) is an equation involving ' 

   x,\ y,\ y',\ y'',\ \dots .

The order of a differential equation is the order n of the highest derivative that appears. If the highest derivative appears only in integer powers, then the degree of the equation is the highest power of the highest derivative.

When a differential equation of order n has the form 

  F\left(x, y, y', y'',\ \dots,\ y^{(n)}\right) = 0

it is called an implicit differential equation whereas the form 

  F\left(x, y, y', y'',\ \dots,\ y^{(n-1)}\right) = y^{(n)}

is called an explicit differential equation.

A differential equation not depending on x is called autonomous, and one with no terms depending only on x is called homogeneous.

A differential equation is said to be linear if only single powers of the dependent variable y and its derivatives appear in the equation.


Ordinary differential equations which can be categorised by three factors: 

   * Linear vs. Non-linear
   * Homogeneous vs. Inhomogenous
   * Constant coefficents vs. variable coefficients

Information below provides methods for the solution of these differing ODEs:

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Homogeneous linear ODEs with constant coefficients

The first method of solving linear ordinary differential equations with constant coefficients is due to Euler, who realized that solutions have the form ezx, for possibly-complex values of z. Thus to solve 

   \frac {d^{n}y} {dx^{n}} + A_{1}\frac {d^{n-1}y} {dx^{n-1}} + \cdots + A_{n}y = 0

we set y = ezx, leading to 

   z^n e^{zx} + A_1 z^{n-1} e^{zx} + \cdots + A_n e^{zx} = 0

so dividing by ezx gives the nth-order polynomial 

   F(z) = z^{n} + A_{1}z^{n-1} + \cdots + A_n = 0

In short the terms 

   \frac {d^{k}y} {dx^{k}}\quad\quad(k = 1, 2, \cdots, n).

of the original differential equation are replaced by zk. Solving the polynomial gives n values ofz, z_1, \dots,z_n. Plugging those values into e^{z_i x} gives a basis for the solution; any linear combination of these basis functions will satisfy the differential equation.

This equation F(z) = 0, is the " characteristic " equation considered later by Monge and Cauchy.


Example 

y''''-2y'''+2y''-2y'+y=0 \,

has the characteristic equation

z^4-2z^3+2z^2-2z+1=0 \,.

This has zeroes, i, −i, and 1 (multiplicity 2). The solution basis is then

e^{ix} ,\, e^{-ix} ,\, e^x ,\, xe^x \,.

This corresponds to the real-valued solution basis

\cos x ,\, \sin x ,\, e^x ,\, xe^x \,.


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Partial differential equation

In mathematics, a partial differential equation(or PDE) is a relation involving an unkown function of several independent variables and its partial derivatives with respect to those variables.

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