Let `y^(prime)(x)+y(x)g^(prime)(x)=g(x)g^(prime)(x),y(0),x in R ,` where `f^(prime)(x)` denotes `(dy(x))/(dx),` and `g(x)` is a given non-constant differentiable function on `R` with `g(0)=g(2)=0.` Then the value of `y(2)` is______

Options

A. 0
B. 1
C. `-1`
D. 2

Answer

A

Explanation

We have,
`(dy(x))/(dx)+g'(x)y(x)=g(x)g'(x)`
`rArr" "y(x)e^(g(x))=e^(g(x)){g(x)-1}+C" …(1)"`
It is given that y(0) = 0 and g(0) = 0
`therefore" "0=e^(0)(0-1)+CrArr C=1`
Putting C = 1 in (i), we get
`y(x)e^(g(x))=e^(g(x)){g(x)-1}+1`
Putting x = 2, we get
`y(2)e^(0)=e^(0)(0-1)+1 rArr y(2)=0" "[because g(0)=0]`