Question

The differential equation \(x \frac{\mathrm{~d} y}{\mathrm{~d} x}=2 y\) represents ________

• A. a family of circles with radius c .
• B. a family of parabolas with vertex at the origin and axis along the positive Y -axis
• C. a family of parabolas with vertex at origin and axis along the positive X -axis
• D. a family of ellipses

Hint:

Separate the variables, integrate, and identify the curve y = c x^2.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Separate the variables in \(x\dfrac{dy}{dx} = 2y\): \(\dfrac{dy}{y} = \dfrac{2\,dx}{x}\).

Step 2: Integrate
\(\log y = 2\log x + \log c\), so \(y = cx^2\).

Step 3: Identify the curve
This can be written \(x^2 = \dfrac{1}{c}y\), a parabola with vertex at the origin.
Its axis is along the Y-axis (positive Y-axis for \(c > 0\)).

Final Answer:
It represents parabolas with vertex at the origin and axis along the positive Y-axis. \[ \boxed{\text{(B) }\text{a family of parabolas with vertex at the origin and axis along the positive Y-axis}} \]