If `f:Rto[0,oo]` be a function satistying the property `f(x+y)-f(x-y)=f(x)[f(y)-f(-y)]Aay, xepsilonR, f^(')(0)=In a, f(0)=1`
The soluton of differential equation `(dy)/(dx)=((log_(a)f(x)+log_(a)f(y))^(2))/([log_(a)f(x)+2][log_(a)f(y)-2])` is

Options

A. `(y-2)/(x+2)-1/2In |1+2((y-2)/(x+2))|=2(n(x+2)+c`
B. `(y+2)/(y-2)-In |1+ (y+2)/(x-2)|=c`
C. `(x+2)(y-2)=In((x+2)/(y-2))+c`
D. none of these

Answer

A

Explanation

`f_(')(0)=lim_(hto 0) (f(h)-1)/h=Ina `
`f^(')(x)=lim_(hto 0) (f(x))/2[(f(h)-1)/h+(f(-h)-1)/h]=(f(x))/2 2 In a`
`(f^(')(x))/(f(x))=In a implies In f (x) =In a^(x)+C`
`f(0)=1impliesC=0impliesf(x)=a^(x)`
`(dy)/(dx)=((x+y)^(2))/((x+2)(y-2))` but (put `x+2=X, y-2=Y`)
`(dY)/(dX)=((X+Y)^(2))/(XY)`
`implies(y-2)/(x+2)-1/2 In |1+(2(y-2))/(x+2)|=2 In (x+2)+C`