The differential equations are classified as:
- Ordinary Differential Equations
- Partial Differential Equations
Ordinary Differential Equation
The “Ordinary Differential Equation” also known as ODE is an equation that contains only one independent variable and one or more of its derivatives with respect to the variable. Thus, the ordinary differential equation is represented as the relation having one independent variable x, the real dependent variable y, with some of its derivatives y’, y”, ….yn,…with respect to x. The ordinary differential equation can be homogenous or non-homogenous
Example: (d2y/dx2) + (dy/dx) = 3y cosx
The above differential equation example is an ordinary differential equation since it does not contain partial derivatives.
Homogenous Differential EquationA differential equation in which the degree of all the terms is the same is known as a homogenous differential equation. In general they can be represented as P(x,y)dx + Q(x,y)dy = 0, where P(x,y) and Q(x,y) are homogeneous functions of the same degree.
Examples of Homogenous Differential Equation:
- y + x(dy/dx) = 0 is a homogenous differential equation of degree 1
- x4 + y4(dy/dx) = 0 is a homogenous differential equation of degree 4
- xy(dy/dx) + y2 + 2x = 0 is not a homogenous differential equation
Non-Homogenous Differential EquationA differential equation in which the degree of all the terms is not the same is known as a homogenous differential equation.
Example: xy(dy/dx) + y2 + 2x = 0 is not a homogenous differential equation.
One of the types of a non-homogenous differential equation is the linear differential equation, similar to the linear equation. The differential equation of the form (dy/dx) + Py = Q (Where P and Q are functions of x) is called a linear differential equation. (dy/dx) + Py = Q (Where P, Q are constant or functions of y). The general solution is y × (I.F.) = ∫Q(I.F.)dx + c where, I.F(integrating factor) = e∫pdx