Let `y^(prime)(x)+y(x)g^(prime)(x)=g(x)g^(prime)(x),y(0) = 0,x in R ,` where `f^(prime)(x)` denotes `(df(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______

Explanation

`y^(')(x)+y(x)g^(')(x)=g(x)g^(')(x)`
or `e^(g(x))(x) + e^(g(x))y(x) = e^(g(x))g(x)g^(')(x)`
`therefore y(x)e^(g(x)) = e^(g(x))g^(')(x)`
`therefore y(x)e^(g(x)) = int e^(g(x))g^(')(x)dx= inte^(x)tdt,` where `g(x)=t`
`=(t-1)e^(t)+c`
`therefore y(x)e^(g(x)) = (g(x)-1)^(e(g(x))+c`
Put `x=0`. Then `0=(0-1).1+c` or `c=1`.
Put `x=2`. Then `y(2).1=(0-1).(1)+1`
`y(2)=0`