Question

$\dfrac{dy}{dx} = -\dfrac{y}{x}$ is a differential equation for a/an

• A. circle
• B. ellipse
• C. bell-shaped curve
• D. hyperbola

Hint:

If a differential equation reduces to \(xy = k\), the curve is always a hyperbola.

Answer 1

The Correct Option is D

Step 1: Solve the differential equation.
\[ \frac{dy}{dx} = -\frac{y}{x} \Rightarrow \frac{dy}{y} = -\frac{dx}{x} \] Integrating: \[ \ln y = -\ln x + C \Rightarrow \ln(xy) = C \Rightarrow xy = C' \] Step 2: Interpretation.
The equation \(xy = \text{constant}\) represents a rectangular hyperbola.
Step 3: Conclusion.
Thus the differential equation represents a hyperbola.