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. \(2, 6\)

Hint:

The order of a differential equation is the highest order of derivative present, and the degree is the highest power of the highest order derivative after the equation is made free of radicals and fractions involving derivatives.

Answer 1

The Correct Option is C

To ascertain the order and degree of the differential equation, execute the following steps:

1. Order: The order corresponds to the highest-order derivative present. For the equation: \[ \left[ 1 + \left(\frac{dy}{dx}\right)^2 \right]^3 = \frac{d^2y}{dx^2}, \] the highest derivative is \(\frac{d^2y}{dx^2}\). 

Therefore, the order of the equation is \(2\). 2. Degree: The degree is the exponent of the highest-order derivative, contingent on the equation being devoid of radicals and fractional derivative powers. 

In this instance, \(\frac{d^2y}{dx^2}\) is raised to the power of one, and no fractional powers of \(\frac{d^2y}{dx^2}\) are present. 

Consequently, the degree of the equation is \(1\). The order and degree of the given differential equation are thus \(2\) and \(1\), respectively. The correct option is (C).