The differential equation $x \frac{dy}{dx} + y = c$, where $c$ is a constant, represents a family of:
Options:
(A) circles
(B) parabolas
(C) ellipses
(D) hyperbolas
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2020 Q.26)**
**1. Intuitive Physical Setup & Given Parameters**:
- First-order differential equation: $x \frac{dy}{dx} + y = c$
**2. Step-by-Step Mathematical Derivation**:
1. **Recognizing the Product Rule**:
Notice that the left hand side is the exact derivative of the product $x y$:
$\frac{d}{dx}(x y) = x \frac{dy}{dx} + y$
Therefore:
$\frac{d}{dx}(x y) = c$
2. **Integrating Both Sides**:
$x y = \int c\,dx = c x + k$
where $k$ is an arbitrary constant of integration.
3. **Rearranging Geometric Equation**:
$y = c + \frac{k}{x} \implies x(y - c) = k$
This is the standard equation of a family of **rectangular hyperbolas** with asymptotes along $x = 0$ and $y = c$.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** circles: ❌ Incorrect. Equation of circles is $x^2 + y^2 = r^2$.
- **(B)** parabolas: ❌ Incorrect. Equation of parabolas is $y = a x^2 + b$.
- **(C)** ellipses: ❌ Incorrect. Equation of ellipses is $x^2/a^2 + y^2/b^2 = 1$.
- **(D)** hyperbolas: ✅ **CORRECT**. $x(y - c) = k$ represents rectangular hyperbolas.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **D**
- **Exam Rule**: $d(xy) = c dx \implies x(y-c) = k$, which represents a family of hyperbolas.
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