Question

The solution of the differential equation $ \frac{dy}{dx} = -\frac{x}{y} $ represents family of:

• A. Parabolas
• B. Circles
• C. Ellipses
• D. Hyperbolas

Hint:

When solving a differential equation that involves $ \frac{dy}{dx} = -\frac{x}{y} $, integrate both sides after rearranging to find a relationship between $x$ and $y$. The resulting equation $ x^2 - y^2 = C' $ represents a hyperbola.

Answer 1

The Correct Option is D

The differential equation is given as: \[ \frac{dy}{dx} = -\frac{x}{y} \] Step 1: Rearrange the equation to separate variables: \[ y \, dy = -x \, dx \] Step 2: Integrate both sides of the equation: \[ \int y \, dy = - \int x \, dx \] Evaluating the integrals yields: \[ \frac{y^2}{2} = - \frac{x^2}{2} + C \] Step 3: Multiply the equation by 2: \[ y^2 = -x^2 + C' \] This can be rearranged to: \[ x^2 + y^2 = C' \] This equation represents a family of circles centered at the origin.