10.6.1 Variables Separable Method

In solving differential equations, separation of variables was introduced initially by Leibniz and later it was formulated by John Bernoulli in the year . A first order differential equation is separable if it can be written as h y y g x ′ = where the left side is a product of ′ y and a function of y and the right side is a function of x . Rewriting a separable differential equation in this form is called the method of separation of variables. Finding a solution to a first order differential equation will be simple if the variables in the equation can be separated.

An equation of the form f x g y dx x g y dy is called an equation with variable separable or simply a separable equation . Rewrite the given differential equation as f x x dx g g y dy = − ...( ) Integration of both sides of ( ) yields the general solution of the given differential equation as x dx g g y dy = − , where C is an arbitrary constant. Remarks . No need to add arbitrary constants on both sides as the two arbitrary constants are combined together as a single arbitrary constant.

. A solution with this arbitrary constant is the general solution of the differential equation. “Solving a differential equation” is also referred to as “integrating a differential equation”, since the process of finding the solution to a differential equation involves integration. Example .

Solve = + y . Given that dx = + y . ... ( ) The given equation is written in the variables separable form = dx ...

( ) Integrating both sides of ( ), we get tan tan C . ... ( ) But tan tan x = tan − xy . ...