Question

The solution of the differential equation $2x\frac{dy}{dx}-y=0;y\left(1\right)=2$ represents _______

• A. ellipse
• B. parabola
• C. circle
• D. straight line

Answer 1

The correct option is B parabola

$2x\frac{dy}{dx}=y\Rightarrow 2\frac{dy}{y}=\frac{dx}{x}$
Integrating both sides,
$2\log y=\log x+c$
Given $y\left(1\right)=2$
$\Rightarrow 2\log 2=\log 1+c\Rightarrow c=\log 4$
Therefore, $\log y^2=\log x+\log 4$
$\Rightarrow y^2=4x\to \text{a parabola}$