The solution of the differential equation
`(1+y^(2)) dx = (tan^(-1) y - x) dy` is

Options

A. ` x = tan ^(-1)y -1+Ce^(-tan^(-1)y)`
B. ` y = tan^(-1)y+1 +Ce^(-tan^(-1)y)`
C. ` x = tan^(-1)y +Ce^(-tan^(-1))y`
D. None of the above

Answer

a

Explanation

The Given differential equation is
`(1+y^(2)) dx = (tan^(-1) y - x) dy`
This equation can be rewritten as
` (dx)/(dy) + x/(1+y^(2)) = (tan^(-1)y)/(1+y^(2))`
This is of the form ,
`(dx)/(dy) +Px = Q`
Here, ` P = 1/(1+y^(2)) and Q = (tan^(-1)y)/(1+y^(2))`
` :. IF = e^(int 1/(1+y^(2))dy) = e^(tan-1y)`
Therefore , required solution is
` xe^(tan-1y) = int e^(tan^(-1)) * (tan^(-1)y)/(1+y^(2)) dy +C`
` rArr xe^(tan-1y) = (tan ^(-1)y-1)e^(tan^(-1)y) +C`
` rArr x = tan^(-1) y - 1 +Ce ^(-tan ^(-1)y)`