Classification of Differential Equation and Examples
1. Ordinary differential equation in which all derivatives are ordinary derivatives of one or more dependent variable with respect to a single independent variable.
2. A partial differential equation is a differential equation containing at least one partial derivative of some dependent variable.
3. The Order of a Differential Equation is the order of the neglect order derivative which appears in the equation.
4. A Differential Equation is linear in a set of one or more of its dependent variables if and only if each learn of the equation which contains a variable of the set or any of their duration is of the first degree in those variable and their derivative.
5. A Differential Equation which is not linear in some dependent variable is said to be nonlinear in that variables. A differential Equation which is not linear in the set of all its dependent variable is simply said to be nonlinear.
Example 1
The Equation 
is a linear differential equation of second order the presence of product xy and the term Cos X does not alter this fact that the equation is linear because by definition linearity is determined so Let y by the way the dependent variable y. and its derivatives enter into combination away themselves within each term of the equation.
Example 2
The Equation 
is a nonlinear equation becomes of the occurrence of the product of y and one of derivatives.
Example 3
The Equation 
is linear in the dependent variable V but nonlinear in the dependent variable a because Sin U is a nonlinear function of U. The equation is also nonlinear
Example 4
The Equation 
is linear in each dependent variable x and y. However because of the term xy is not linear in the set of dependent variable (x, y) or consequence the equation is also nonlinear.
Example 5
The equation 3x^2 dx + (Sin x) y = 0 is neither linear or nonlinear division by dx transform it into the equation 3x^2 + (Sin x) y = 0 which is linear in y but division by dy gives 3x^2 dx/dy + Sin x = 0