Let $y'(x) + y(x)\,g'(x) = g(x)\,g'(x)$, $y(0) = 0$, $x \in R$, where $f'(x)$ denotes $\frac{df(x)}{dx}$ 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

$y' + y\,g' = g\,g',\qquad y(0)=0$ $g(0)=g(2)=0,\quad g \text{ non-constant differentiable}$ $\text{To find: } y(2)$ $\text{Plan: multiply by } e^{g(x)} \Rightarrow (y\,e^{g})' = g\,g'\,e^{g}$ $\text{IF} = e^{\int g'\,dx} = e^{g(x)}$ $e^{g}y' + e^{g}g'y = g\,g'\,e^{g} \Rightarrow (y\,e^{g})' = g\,g'\,e^{g}$ $\dfrac{d}{dx}\left(y\,e^{g}\right) = g\,g'\,e^{g}$ $\left[y\,e^{g}\right]_0^2 = y(2)e^{g(2)} - y(0)e^{g(0)}$ $= y(2)\cdot 1 - 0 = y(2)$ $\int_0^2 g\,g'\,e^{g}\,dx,\qquad u=g(x)\Rightarrow du=g'\,dx$ $= \int_{g(0)}^{g(2)} u\,e^{u}\,du = \int_0^0 u\,e^{u}\,du = 0$ $y(2) = 0$ $\boxed{y(2) = 0}$