If an equation involves one dependent variable and its derivatives w.r.t. one (or) more independent variables then it is called a Differential Equation.
A differential equation is said to be an ordinary differential equation if it contains only one independent variable.

A differential equation is said to be a partial differential equation if it contains atleast two independent variables.

                         
In this chapter we deal with the solving of first order differential equations in which
(1) Variables are separable type
(2) Homogeneous equations type
(3) Non-homogeneous equations type
(4) Linear equations type

(1) Variables Separable Type
If a D.E. can be expressed in the form    (or) f(x) dx  =  g(y)dy
(or)  f(x)dx  −  g(y)dy  =  0 then it is said to be of type variables separable.
Sometimes,  
can't be reduced directly to the variables separable form. In this case using some substitution we reduce the equation to the variables separable form.
 

(2) Homogeneous Equations type
An equation of the form     
      where f(x, y) and g(x, y) are homogeneous functions of x, y and of the same degree is called a homogeneous differential equation of first order.
Such an equation can be reduced to the form

Using the substitutions  y  =  vx (v is a variable)  and   

The equation is transformed into variables separable form in v and x and hence can be solved.
The solving of D.E. naturally involves integration.
 

(3) Non-Homogeneous Equation Type
       An equation of the form    where a, b, c, a', b', c' are constants is called a non-homogeneous differential equation of first order.

This equation can be reduced to Variables separable form by substituting ax  +  by  =  V.

Hence it can be solved.

Now choose h and k such that ah  +  bk  +  c  =  0
and a'h  +  b'k  +  c'  =  0
Then the differential equation becomes
  which is homogeneous.

Now this equation can be solved by substituting Y = VX
Finally by replacing X by x - h and Y by y - k we shall get the solution in x and y.
Similarly a linear differential equation of first order in x is of the form
   where P, Q are functions of Y.
In this case integrating factor is e ∫Pdy and solution is x(I.F.) = ∫ Q (I.F.) dy + C.
 

(4) Linear Equation Type
An equation of the form     where P, Q are functions of x is called a linear differential equation of first order in y.
Multiplying the equation    by e ∫Pdy we get
  . e ∫Pdx  +  y. P e ∫Pdx  =  Q . e ∫Pdx on integrating
        y. e ∫Pdx  =  ∫ Q . e ∫Pdx dx  +  c  which is the required solution.
Here e ∫Pdx is called the integrating factor of the D.E.