Separable differential equations Find the general solution of the following equations.

, where y > 0

Problem 24E

Separable differential equations Find the general solution of the following equations.

, where y > 0

Answer;

Step-1;

DEFINITION : A differential equation is said to be of type “variable separable” if it can be expressed in such a way , so that the coefficient of dx is a function of of x alone and the coefficient of dy is a function of y alone.

The general form of such differential equation can be written as

f(x) dx = g(y)dy ……………….(1)

Integrating both sides and adding an arbitrary constant C , we get the general solution as

f(x)dx = +C

Working rule of solving by the method of separation of variables;

Write the given differential equation in the formf(x) dx = g(y)dy

That is make the coefficient of dx as an expression of x alone and that of dy as an expression of y alone

2. Integrate both sides and add an arbitrary constant to any one side and get the general solution.

Step-2;

In this problem we have to check whether the given equation is separable or not and if they are separable we have to solve the given initial value problem.

Given equation is ; = y (+1) , where y >0

dy = (………………(2)

From (1) , it is in the form of f( x) dx = g(y) dy, where f(x) = , and g(y) = .

That is , the coefficient of dx is a function of of ‘x’ alone and the coefficient of dy is a function of ‘y’ alone.

Thus , by the definition the given equation is separable.

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