Question

The differential equation representing the family of curves $y^2 = a(ax + b)$ where $a$ and $b$ are arbitrary constants, is of:

• A. order 1, degree 1
• B. order 1, degree 3
• C. order 2, degree 3
• D. order 1, degree 4
• E. order 2, degree 1

Hint:

Always reduce equation to standard differential form before identifying degree.

Answer 1

The Correct Option is C

Concept:
• Order = Number of independent arbitrary constants.
• Degree = Power of highest order derivative (after removing radicals/fractions).

Step 1: Determine the order.
Given: \[ y^2 = a^2 x + ab \] Let \( A = a^2 \), \( B = ab \) \[ y^2 = Ax + B \] Since \( A \) and \( B \) are independent constants, \[ \text{Order} = 2 \]

Step 2: Differentiate to eliminate constants.
Differentiate once: \[ 2y \frac{dy}{dx} = A \quad \cdots (1) \] Differentiate again: \[ 2\left[ y \frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 \right] = 0 \] \[ \Rightarrow y y'' + (y')^2 = 0 \]

Step 3: Determine degree.
Highest order derivative is \( y'' \) and its power is 1. \[ \text{Degree} = 1 \]

Step 4: Final answer.
\[ \boxed{\text{Order = 2, Degree = 1}} \]