Question

The order and degree of the differential equation: \[ \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = \frac{d^2y}{dx^2}, \] respectively, are:

• A. 1, 2
• B. 2, 3
• C. 2, 1
• D. 2, 6
• E.

Hint:

The order of a differential equation is determined by the highest derivative present, while the degree is determined by the power of the highest order derivative after eliminating radicals and fractions involving derivatives.

Answer 1

The Correct Option is C

To determine the order and degree of the given differential equation, follow these steps: 1. Order: The order of a differential equation is the highest order derivative present in the equation. In this case, the highest order derivative is: \[ \frac{d^2y}{dx^2}. \] Therefore, the order of the equation is \( 2 \). 2. Degree: The degree of a differential equation is the power of the highest order derivative, provided the equation is free from fractional powers and radicals with respect to derivatives. Here, the equation is: \[ \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = \frac{d^2y}{dx^2}. \] The highest order derivative \( \frac{d^2y}{dx^2} \) is raised to the power of \( 1 \). Thus, the degree of the equation is \( 1 \). Hence, the order and degree of the given differential equation are \( 2 \) and \( 1 \), respectively. Final Answer: (C) 2, 1.