Question

If \( p \) and \( q \) are respectively the order and degree of the differential equation \( \frac{d}{dx} \left( \frac{dy}{dx} \right)^3 = 0 \), then \( (p - q) \) is:

• A. \( 0 \)
• B. \( 1 \)
• C. \( 2 \)
• D. \( 3 \)

Hint:

The order of a differential equation is determined by the highest derivative, and the degree is the highest power of the dependent variable.

Answer 1

The Correct Option is C

Step 1: Finding the order and degree.
The given differential equation is \( \frac{d}{dx} \left( \frac{d}{dx} y^3 \right) = 0 \). First, let's find the order and degree of this equation. The function is \( y^3 \), so we have: \[ \frac{d}{dx} y^3 = 3y^2 \frac{dy}{dx} \] Now, applying the derivative again: \[ \frac{d}{dx} \left( 3y^2 \frac{dy}{dx} \right) = 6y \left( \frac{dy}{dx} \right)^2 + 3y^2 \frac{d^2y}{dx^2} \] This is a second-order differential equation, so the order is \( 2 \). Since the highest power of \( y \) is 3, the degree is \( 3 \). Thus, \( p = 2 \) and \( q = 3 \), so \( p - q = 2 - 3 = -1 \).