Question

The solution of the differential equation \[2x\frac{dy}{dx} - y = 3\] represents

Options

• circles

• straight lines

• ellipses

• parabolas

Answer 1

parabolas

 

We have,

\[2x\frac{dy}{dx} - y = 3\]

\[ \Rightarrow 2x\frac{dy}{dx} = 3 + y\]

\[ \Rightarrow \frac{1}{3 + y}dy = \frac{1}{2x}dx\]

Integrating both sides, we get

\[\int\frac{1}{3 + y}dy = \frac{1}{2}\int\frac{1}{x}dx\]

\[ \Rightarrow \log \left| 3 + y \right| = \frac{1}{2}\log \left| x \right| + \log C\]

\[ \Rightarrow \log \left| 3 + y \right| - \log \left| x^\frac{1}{2} \right| = \log C\]

\[ \Rightarrow \log \left| \frac{3 + y}{\sqrt{x}} \right| = \log C\]

\[ \Rightarrow \frac{3 + y}{\sqrt{x}} = C\]

\[ \Rightarrow 3 + y = C\sqrt{x}\]

Squaring both sides, we get

\[ \left( 3 + y \right)^2 = Cx . . . . . \left( 1 \right)\]

\[\text{ Thus, }\left( 1 \right)\text{ represents the equation of parabolas .}\]