The solution of differential equation 2xdy/dx-y = 3 represents :

The correct answer is Option (3) → Parabolas

$2x\frac{dy}{dx}-y=3⇒\frac{dy}{dx}-\frac{y}{2x}=\frac{3}{2x}$

$I.F.=e^{\int-\frac{1}{2x}dx}=e^{-\frac{1}{2}\log x}=\frac{1}{\sqrt{x}}$

multiplying eq. with I.F. and integrating wrt x

$\int\frac{1}{\sqrt{x}}\frac{dy}{dx}-\frac{ydx}{2x\sqrt{x}}=\int\frac{3}{2x^{3/2}}dx$

$=\frac{y}{\sqrt{x}}=\frac{3}{2}\frac{x^{-\frac{1}{2}}}{-\frac{1}{2}}+c$

$=\frac{y}{\sqrt{x}}=-\frac{-3}{\sqrt{x}}+c$

so $y=-3+c\sqrt{x}$ depicts parabola