In the field of physical and engineering sciences differential equation is plays a crucial role. Also, several problems arising in bioscience, ecology, and economics can be formulated in the form of differential equation. For investigation of different real world phenomena such as chemical reactions, heat conduction, current flows in electrical circuits, bacteria population growth, study planetary motion etc. understanding , formulating, and solving differential equations is required.
Definition: A differential equation is a relation between independent variables, dependent variables and their differential coefficients of any order is called a differential equation.
A differential equations are classified into two types (i) Ordinary Differential Equations and (ii) Partial Differential Equations.
Ordinary Differential Equation:
A differential equation which involves ordinary derivatives of one or more than one dependent variables with respect to a single independent variable is called Ordinary Differential Equation (ODE).
Examples:
$(i) \dfrac{dy}{dx}+2\displaystyle\frac{y}{x}=x^2$
$(ii) \dfrac{d^2y}{dx^2}+7\dfrac{dy}{dx}+6y=\sin{x} $
$(iii) \dfrac{d^3y}{dx^3}+x\displaystyle(\dfrac{dy}{dx})^3+5y=x\sin{x}$
$(iv) \dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}-3y=e^{x}$
$(v) \dfrac{d^4y}{dx^4}+(\dfrac{d^3y}{dx^3})^2+(\dfrac{dy}{dx})^5-3y=e^{x}$
$(vi) (\dfrac{dy}{dx})^3-5y=x^2e^{3x}$
Classification of an Ordinary Differential Equation: Order and degree of differential equation
To classify an ordinary differential equation mainly check order and degree of the given differential equation.
Order of a Differential Equation: The order of a differential equation is the order of the highest derivatives appearing in the differential equation.
Example: The ordinary differential equation $(i)$ and $(vi)$ are of first order. Equations $(ii)$ and $(iv)$ are of second order, equation $(iii)$ and $(v)$ are of third and fourth order respectively.
Degree of an ordinary differential Equation: The degree of an ordinary differential equation is the degree or power of the highest order derivative term, when the equation is free from radicals and fractions in respect of the derivative terms.
Example: The ordinary differential equation $(i), (ii), (iii), (iv) $ and $(v)$ are of degree one where as equation $(vi)$ is of degree three .