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 ascertain the order and degree of the provided differential equation, execute the following steps:1. Order: The order is defined by the highest-order derivative present. In this instance, the highest-order derivative is \( \frac{d^2y}{dx^2} \). Consequently, the order of the equation is \( 2 \).2. Degree: The degree is determined by the exponent of the highest-order derivative, assuming the equation is free of fractional powers and radicals concerning derivatives. The given 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 \( 1 \). Therefore, the degree of the equation is \( 1 \).In conclusion, the order and degree of the differential equation are \( 2 \) and \( 1 \), respectively. Final Answer: (C) 2, 1.