Solutions of Differential Equation

A solution of a differential equation with arbitrary constant equal in number to the order if differential equation  is called General Solution .

Other solutions obtained by  giving particular values to the arbitrary constant in the general solution are called Particular Solutions .

Also we know that the general solution of a function contains an arbitrary constant . Therefore the solution of differential equations , resulting as they do from the operations of integration , must contain arbitrary constants, equal in number to the number of time the integration is involved in obtaining the solution which is equal to the order of differential equation .

Thus we see that the most general solution of a differential equation of the   nth   order must contain       and only    n    independent arbitrary constants .

An ordinary differential equation  is a differential equation in which the unknown function is a function of a single independent variable. In the simplest form, the unknown function is a real or complex valued function, but more generally, it may be vector-valued or matrix-valued: this corresponds to considering a system of ordinary differential equations for a single function.

Ordinary differential equations are further classified according to the order of the highest derivative of the dependent variable with respect to the independent variable appearing in the equation. 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 
 is a second-order differential equation. In the classical literature a distinction is also made between differential equations explicitly solved with respect to the highest derivative and differential equations in an implicit form. Also important is the degree, or (highest) power, of the highest derivative in the equation . A differential equation is called a nonlinear differential equation if its degree is not one.
A partial differential equation  is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives. 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.


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