Question

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

• A. Circle
• B. Parabola
• C. Straight-line
• D. Ellipse

Answer 1

Answer: (B)

Parabola

$2x\frac{dy}{dx}-y=0$

$⟹2\frac{dy}{y}=\frac{dx}{x}$

$⟹2\int \frac{dy}{y}=\int \frac{dx}{x}$

$⟹2lny=ln\left(x\right)+C$

Using $y\left(1\right)=2$, we get

$⟹2ln\left(2\right)=ln\left(1\right)+C$

$⟹C=ln\left(4\right)$

Therefore, the equation becomes

$2lny=ln\left(x\right)+ln\left(4\right)$

$⟹y^2=4x$

This is the equation of a parabola.

Hence, the answer is option (B).