Let y'(x) + y(x)g'(x) = g(x)g'(x), y(0) = 0, x ∈ R, where f'(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:

dy/dx + g'(x)y = g(x)g'(x)
I.F. = e∫g'(x)dx = eg(x)
General solution is y eg(x) = ∫g'(x)g(x)eg(x)dx + C
I = ∫g'(x)g(x)eg(x)dx
Substitute eg(x) = t
I = ∫t dt = tln t - t = eg(x)(g(x) - 1)
So, y eg(x) = eg(x)(g(x) - 1) + C
Given y(0) = 0 ⇒ C = 1
For y(2), put x = 2 and g(2) = 0
y(2) = -1 + 1 = 0