The solution of the differential equation 2 x d y d x - y = 4 represents a family of

Given differential equation is $2x\left(\frac{dy}{dx}\right)-y=4.$.

$\Rightarrow 2x\frac{dy}{dx}=4+y$

$\Rightarrow 2\frac{dy}{4+y}=\frac{dx}{x}$

On integrating both sides, we get 

$\Rightarrow \int \frac{2}{4+y}dy=\int \frac{1}{x}dx$

$\Rightarrow 2\log \left|\left(4+y\right)\right|=\log \left|x\right|+\log \left|c\right|$

Using $\log a+\log b=\log \left(ab\right)$ and $\log a^n=n\log a,$

$\Rightarrow \log {\left(4+y\right)}^2=\log \left|xc\right|$

$\Rightarrow {\left(y+3\right)}^2=\left|cx\right|$, which is of the form ${\left(y-k\right)}^2=4a\left(x-h\right)$ and this is the general equation of a parabola.

Thus, the equation ${\left(y+3\right)}^2=\left|cx\right|$, represents a family of  parabolas.