$\Rightarrow x y=2 x+\frac{d y}{d x}$
$\Rightarrow x(y-2)=\frac{d y}{d x}$
$\Rightarrow \int x d x=\int \frac{d y}{y-2}$
$\Rightarrow \frac{x^{2}}{2}=\ln |y-2|+\ln c$
$\Rightarrow e^{\frac{x^{2}}{2}}=c(y-2) \quad-(2)$
Now, put $x=a$ in eqn ( 1$)$
$\Rightarrow y=-a^{2}$
So, by eqn ( 2$)$ $\therefore e^{\frac{a^{2}}{2}}=c\left(-a^{2}-2\right)$
$\Rightarrow c=-\frac{e^{\frac{a^{2}}{2}}}{\left(a^{2}+2\right)}$
Put this value in (2) $\therefore e^{\frac{x^{2}}{2}}=-\frac{e^{\frac{a^{2}}{2}}}{\left(a^{2}+2\right)}(y-2)$
$\Rightarrow-y+2=\left(a^{2}+2\right) e^{\frac{x^{2}-a^{2}}{2}}$
$\Rightarrow y=2-\left(2+a^{2}\right) e^{\frac{x^{2}-a^{2}}{2}}$
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