Question

Let $y\left(x\right)+y\left(x\right)g^{\prime }\left(x\right)=g\left(x\right)g^{\prime }\left(x\right),y\left(0\right)=0,xϵ,\text{where,}{f}^{\prime }\left(x\right)\text{denotes}\frac{df\left(x\right)}{dx}\text{and}g\left(x\right)$ is a given non-constant differentiable function on with $g\left(0\right)=g\left(2\right)=0.$ Then the value of $y\left(2\right)$ is

Answer 1

$y^{\prime }x+y\left(x\right)g^{\prime }\left(x\right)=g\left(x\right).g^{\prime }\left(x\right)$
$\Rightarrow$ which is linear differential equation
I.F. $=E^{\int g^{\prime }\left(x\right)dx}=e^{g\left(x\right)}$
Solution is
$y\left(x\right)e^{g\left(x\right)}=\int e^{g\left(x\right)}.g\left(x\right)g^{\prime }\left(x\right)dx$
$y\left(x\right)e^{gx}=e^{g\left(x\right)}\left(g\right(x)-1)+k$
where k is a constant of integration
For $x=0,k=1$
For $x=2,y\left(2\right)=0$