Question

The solution of the differential equation $2x\frac{dy}{dx}-y=3$ represents
(a) circles
(b) straight lines
(c) ellipses
(d) parabolas

Answer 1

(d) parabolas

We have,
$$\begin{gathered} 2x\frac{dy}{dx}-y=3 \\ \Rightarrow 2x\frac{dy}{dx}=3+y \\ \Rightarrow \frac{1}{3+y}dy=\frac{1}{2x}dx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \int \frac{1}{3+y}dy=\frac{1}{2}\int \frac{1}{x}dx \\ \Rightarrow \log \left|3+y\right|=\frac{1}{2}\log \left|x\right|+\log C \\ \Rightarrow \log \left|3+y\right|-\log \left|x^{\frac{1}{2}}\right|=\log C \\ \Rightarrow \log \left|\frac{3+y}{\sqrt{x}}\right|=\log C \\ \Rightarrow \frac{3+y}{\sqrt{x}}=C \\ \Rightarrow 3+y=C\sqrt{x} \\ \mathrm{Squaring}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ {\left(3+y\right)}^2=Cx.....\left(1\right) \\ \mathrm{Thus},\left(1\right)\mathrm{represents}\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{parabolas}. \end{gathered}$$