Mathematics - Differential Equation Question with Solution | TestHub

$\frac{dy}{dx}+y\cdot g^{\prime }(x)=g(x)g^{\prime }(x)$ $$\mathrm{IF}=e^{\int g^{\prime }(x)dx}=e^{g(x)}$$ $\therefore$ Solution is $$y\left(e^{g(x)}\right)=\int g(x)\cdot g^{\prime }(x)\cdot e^{g(x)}dx+C$$ Put $g(x)=t,g^{\prime }(x)dx=dt$ $$\begin{array}{rl}& y\left(e^{g(x)}\right)=\int t\cdot e^{t}dt+C\\ =& t\cdot e^t-\int 1\cdot e^{t}dt+C=t\cdot e^t-e^t+C\\ & y{e}^{g(x)}=(g(x)-1)e^{g(x)}+C\dots (\mathrm{i})\end{array}$$ Given, $y(0)=0,g(0)=g(2)=0$ $\therefore$ Eq. (i) becomes, $$\begin{array}{rl}& y(0)\cdot e^{g(0)}=(g(0)-1)\cdot e^{g(0)}+C\\ & \Rightarrow 0=(-1)\cdot 1+C\Rightarrow C=1\\ & \therefore y(x)\cdot e^{g(x)}=(g(x)-1)e^{g(x)}+1\\ & \Rightarrow y(2)\cdot e^{g(2)}=(g(2)-1)e^{g(2)}+1\\ & \text{ where, }g(2)=0\\ & \Rightarrow y(2)\cdot 1=(-1)\cdot 1+1\Rightarrow y(2)=0\end{array}$$