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______

Answer

0

Explanation

`y/(x)+y(x)g'(x)=g(x)g'(x)`
or`e^(g(x))y'(x)+e^(g(x))g(x)y(x)=e^(g(x))g(x)g'(x)`
or `d/(dx)(y(x)e^(g(x)))=e^(g(x))g(x)g'(x)`
`:. y(x)e^(g(x))=inte^(g(x))g(x)g'(x)dx`
`=inte^(t)tdt`, where `g(x)=1`
`=(t-1)e^(t)+c`
`:. y(x)e^(g(X))=(g(x)-1)e^(g(x))+c`
`x=0implies0=(0-1).1+c` or `c=1`
`x=2impliesy(2).1=(0-1).(1)+1`
`:. y(2)=0`