Method of separation of variables

METHOD OF SEPARATION OF VARIABLES

A powerful method of obtaining particular solutions of a p.d.e. is known as separation of variables or product method.

Let z be the dependent variable and x, y be the independent variables in given p.d.e. We assume a solution that z = X (x) Y (y) where X is a function of x alone and Y is a function of y alone, substituting this value of z in the p.d.e. it is possible to write the resulting equations, so that, one side depends on x alone and the other on y alone.

In this way, we can get ordinary differential equations involving X and Y and their derivatives. X and Y can be found and then z = X(x) Y(y)

Problems on method of separation of variables

Example 3.2.1: Solve by using method of separation of variables the equation

Solution :

differentiate (2) partially, we get

where A and k are arbitrary constants.

Example 3.2.2 Using the method of separation of variables solve

Solution: 

EXERCISES 3.2