Question

The solution \( y(x) \) of the differential equation \( y\frac{dy}{dx} + 3x = 0 \), \( y(1) = 0 \), is described by:

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

Hint:

For equations of the form \( y \frac{dy}{dx} + kx = 0 \), integrate directly to get conic forms. Apply conditions to identify whether it represents a circle, ellipse, or parabola.

Answer 1

The Correct Option is A

Step 1: Separate the variables.
Given: \[ y \frac{dy}{dx} + 3x = 0 \Rightarrow y \frac{dy}{dx} = -3x. \] Step 2: Integrate both sides.
\[ \int y\,dy = \int -3x\,dx \Rightarrow \frac{y^2}{2} = -\frac{3x^2}{2} + C. \] Step 3: Apply the initial condition \( y(1) = 0 \).
\[ 0 = -\frac{3(1)^2}{2} + C \Rightarrow C = \frac{3}{2}. \] Step 4: Substitute back.
\[ \frac{y^2}{2} = -\frac{3x^2}{2} + \frac{3}{2} \Rightarrow y^2 + 3x^2 = 3. \] Step 5: Interpret the result.
This represents an ellipse, but since the coefficients differ in magnitude, the form simplifies for equal radii scaling to a circle when constants align symmetrically. Hence, it’s a circle.
Step 6: Final Answer.
The curve described by the equation is a circle.