Differential Equations

If x=f(y) is the solution of the differential equation (1+y2 )+ (x-2e -1y ) dydx=0, y (- 2, 2 ) with f(0)=1, then f ( 1 3 ) is equal to :

1. e /4
2. e /12
3. e /6
4. e /3

Answer

(3) e /6

Solution

Treat x as the dependent variable: dxdy+ x1+y2= 2e -1y1+y2, I.F. =e -1y

Put t= -1y: x et= 2e2tdt=e2t+c

y=0, x=1 c=0 x=e -1y

y= 1 3 f=e /6

Answer: (3)