Definition: An equation involving partial derivatives is called partial differential equation. Shortly it is written as P.D.E.

Some notation for derivatives:

Formulation of partial differential equation:

(i) Elimination of arbitrary constant and

(ii) Elimination of arbitrary function.

Procedure of elimination of arbitrary constant:

Let us consider the relation

g(x,y,z,a,b)=0 ———————————-(i)

where z=ƒ(x,y) , a and b are two arbitrary constants.

We have to eliminate a and b.

The process is that the derivative should be taken of relation (i) with respect to x and y.

Now differentiating (i), partially with respect to x, we get

Similarly, taken partial derivative of (i) with respect to y, we have

From equation (ii) and (iii) we see that they are free from the arbitrary constant.

Hence equations (ii) and (iii) are the required partial differential equations which are obtained by the method of elimination of arbitrary constants.

Example: Obtain a partial differential equation eliminating a and b from the relation

where z is a function of x and y.

Solution: We have ,

Differentiating (i) partially with respect to x and y,we get

Or, zp=ax Or, zq= by

Putting the values of a and b in (i) we get

Which is the required partial differential equation.