Notes On Variables Separable Method - CBSE Class 12 Maths

Method of solution to solve first order, first degree differential equations.

Consider the differential equation dy/dx = F(x,y)

If F(x, y) can be split into two functions, then the variable separable method is applied.

Ex:

Let dy/dx = (1+y2)/ ((1+x)xy)

= 1/x(1+x) . (1+y2)/ y

Here the first function is a function of x, and the other one is a function of y.

The right hand side is expressed as functions of x and y, so that the variables separable method can be applied.

⇒ y/(1+y2) dy/dx = 1/x(1+x)

⇒ y/(1+y2) dy = 1/x(1+x) dx

⇒ y/(1+y2) dy = (1/x - 1/(1+x)) dx (Integrating both the sides )

⇒ ∫ y/(1+y2) dy = ∫ (1/x - 1/(1+x)) dx

⇒ 1/2 ∫ 2y/(1+y2) dy = ∫ (1/x - 1/(1+x)) dx (Using formula ∫ f'(x)/f(x) dx = log f(x) + C)

⇒ 1/2 log(1+y2) = log x - log(1+x) + log C

⇒ log(1+y2)1/2 = log x - log(1+x) + log C

⇒ log(1+y2)1/2 = log Cx/(1+x)

(1+y2)1/2 = Cx/(1+x)

(1+y2) = (Cx/(1+x))2

(1+y2)(1+x)2 = Cx2

Which is the solution of the given differential equation.

Summary

Method of solution to solve first order, first degree differential equations.

Consider the differential equation dy/dx = F(x,y)

If F(x, y) can be split into two functions, then the variable separable method is applied.

Ex:

Let dy/dx = (1+y2)/ ((1+x)xy)

= 1/x(1+x) . (1+y2)/ y

Here the first function is a function of x, and the other one is a function of y.

The right hand side is expressed as functions of x and y, so that the variables separable method can be applied.

⇒ y/(1+y2) dy/dx = 1/x(1+x)

⇒ y/(1+y2) dy = 1/x(1+x) dx

⇒ y/(1+y2) dy = (1/x - 1/(1+x)) dx (Integrating both the sides )

⇒ ∫ y/(1+y2) dy = ∫ (1/x - 1/(1+x)) dx

⇒ 1/2 ∫ 2y/(1+y2) dy = ∫ (1/x - 1/(1+x)) dx (Using formula ∫ f'(x)/f(x) dx = log f(x) + C)

⇒ 1/2 log(1+y2) = log x - log(1+x) + log C

⇒ log(1+y2)1/2 = log x - log(1+x) + log C

⇒ log(1+y2)1/2 = log Cx/(1+x)

(1+y2)1/2 = Cx/(1+x)

(1+y2) = (Cx/(1+x))2

(1+y2)(1+x)2 = Cx2

Which is the solution of the given differential equation.

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