Solve the differential equation:(i) `(1+y^2) + ( x - e^(tan^-1 y) ) dy/dx = 0`(ii) ` x dy/dx + cos^2 y = tan y dy/dx`

Solve the differential equation:(i) `(1+y^2) + ( x - e^(tan^-1 y) ) dy

1.	Solve the differential equation:(i) `(1+y^2) + ( x - e^(tan^-1 y) ) dy/dx = 0`(ii) ` x dy/dx + cos^2 y = tan y dy/dx`
Answer» Correct Answer - ` x e ^(tan^(-1)y) = tan ^(-1) y ` ` (1+ y ^(2)) dx = ( e ^(-tan ^(-1)y ) - x ) dy rArr (dx)/(dy)= ( e^(-tan ^(-1) y) - x )/( ( 1+ y ^(2)))`. ` therefore (dx )/(dy) + (1)/((1 + y ^(2))) * x = ( e ^(-tan ^(-1) y ))/( ( 1 + y ^(2)))` `IF = e ^(int (1)/((1 + y ^(2))) dy) = e ^(tan ^(-1) y ). ` ` therefore x xx e ^(tan^(-1) y ) = int {( e ^(-tan ^(-1) y ))/( ( 1+y ^(2))) xx e ^(tan ^(-1) y )} dy = C = int (1)/(( 1+y ^(2))) dy +C ` ` rArr x xx e^(tan ^(-1)y ) = tan ^(-1) y +C `. Putting ` x = 0 and y = 0` in (i), we get C =0. `" " ` ... (i)