. . 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 . ... ( ) - - Ordinary Differential Equations Using ( ) in ( ) leads to tan − xy = C , which implies y tan say . Thus, y = a + gives the required solution.
Example . Find the particular solution of x ydx satisfying the condition y ( ) Given that ( x dy x ydx = . The above equation is written as dy x dx −+ = . Integrating both sides gives log log( = C , which implies, log( = log C .
Thus, 3log y = log( C , which reduces to log y = log Hence, y ) gives the general solution of the given differential equation. It is given that when x . Then = C (