# Differential Equation

FIRST ORDER DIFFERENTIAL EQUATION
Basic Concepts and Ideas

Differential Equation: – An equation involving one dependent variable and its derivatives with respect to one or more independent variable is called as differential equation.
Example.  , Ordinary differential equation: – An ordinary differential equation is one in which there is only one independent variable, so that all the derivatives occurring in it are ordinary derivatives.
Ex. Partial differential equation: – A Partial differential equation is one involving more than one independent variable, so that all the derivatives occurring in it are partial derivatives.
Example. Order of differential equation: – The order of the differential equation is the order of the highest derivatives present.
Example. (First order) (First order) (Second order)

Degree of differential equation: – The degree of differential equation is the exponent of highest order derivatives present in a differential equation after the differential equation has been rationalised and made free of fractional powers with regards to all derivatives present.
Ex. (Degree 1) (Degree 2) (Degree 1)
First order differential equation: -An equation contains , and function of is called as first order differential equation.
Ex. Solution of differential equation: – Any function free from derivatives, satisfying the differential equation identically is said to be a solution of differential equation.
Example. has a solution  Differentiating with respect to   which is a solution of  order ordinary differential equation: –
An order differential equation in general form is

(1) Solution of (1) containing arbitrary constant is called as general solution.
Any solution obtained from the general solution by giving a particular value to the arbitrary constants is known as particular solution.

Initial Value problem: – A differential equation together with an initial condition is known as initial value problem, which can be expressed in the form of: – It means at x Singular solution: -A differential equation may sometimes have an additional solution that cannot be obtained from the general solution and is then called a singular solution.

1.3 SEPARABLE DIFFERENTIAL EQUATION

Take the first order differential equation (1)
If Equation (1) can be reduced in the form of (i.e. if we can separate the variables in x and y) Then its solution of equation (1) is