The solution of the differential equation $ \frac{dy}{dx} = -\frac{x}{y} $ represents family of:
Show 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.
We are given the differential equation:
\[
\frac{dy}{dx} = -\frac{x}{y}
\]
Step 1: Rearrange the equation:
\[
y \, dy = -x \, dx
\]
Step 2: Integrate both sides:
\[
\int y \, dy = - \int x \, dx
\]
The integrals give:
\[
\frac{y^2}{2} = - \frac{x^2}{2} + C
\]
Step 3: Multiply through by 2:
\[
y^2 = -x^2 + C'
\]
which can be rewritten as:
\[
x^2 - y^2 = C'
\]
This is the equation of a hyperbola. Therefore, the solution of the differential equation represents a family of hyperbolas.
Thus, the correct answer is (D) Hyperbolas.
