The solution of initial value problem $x{y}^{\mathrm{\prime }}+y$

$= 0, y\left(2\right) = – 2$ is given by

$x\frac{\mathrm{d}y}{\mathrm{d}x}=-y\Rightarrow \frac{\mathrm{d}y}{y}+\frac{\mathrm{d}x}{x}=0$

$\begin{array}{l}\Rightarrow \log y+\log x=\text{ Const }\\ \Rightarrow xy=\text{ Const }\end{array}$

Since $y\left(2\right) = – 2$ so Const = 2(–2) = –4

Thus $xy + 4 = 0$ is the required solution.