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. \( 2 \)
• C. \( 1 \)
• 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

To determine the solution, we must ascertain the order and degree of the provided differential equation:

\( \frac{d}{dx} \left( \left( \frac{dy}{dx} \right)^3 \right) = 0 \)

1. Equation Simplification:
The equation is simplified as follows:
Let \( u = \left( \frac{dy}{dx} \right)^3 \). The equation transforms to:
\( \frac{du}{dx} = 0 \)
This implies \( u \) is a constant, leading to:
\( \left( \frac{dy}{dx} \right)^3 = C \), where \( C \) represents a constant.

The given equation is:
\( \frac{d}{dx} \left( \left( \frac{dy}{dx} \right)^3 \right) = 0 \)
We examine the highest order derivative present in the equation as written. The derivative operates on a power of \( \frac{dy}{dx} \), which, upon expansion, incorporates a second derivative.

Differentiating yields: \( \frac{d}{dx} \left( \left( \frac{dy}{dx} \right)^3 \right) = 3 \left( \frac{dy}{dx} \right)^2 \cdot \frac{d^2y}{dx^2} \)
The highest derivative is \( \frac{d^2y}{dx^2} \). Consequently:

Order (p) = 2
The equation is polynomial with respect to its highest order derivative (linear in \( \frac{d^2y}{dx^2} \)). Therefore:

Degree (q) = 1

2. Final Computation:
\( p - q = 2 - 1 = 1 \)

Final Result:
The calculated value of \( p - q \) is 1.