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 determine the order and degree of the differential equation, follow these steps: 

1. Order: The order of a differential equation is the highest order derivative present in the equation. In the given equation: \[ \left[ 1 + \left(\frac{dy}{dx}\right)^2 \right]^3 = \frac{d^2y}{dx^2}, \] the highest derivative is \(\frac{d^2y}{dx^2}\). 

Thus, the order of the equation is \(2\). 2. Degree: The degree of a differential equation is defined as the power of the highest order derivative, provided the equation is free from radicals and fractional powers of the derivatives. 

In this case, \(\frac{d^2y}{dx^2}\) appears to the first power, and there are no fractional powers of \(\frac{d^2y}{dx^2}\) in the equation. 

Thus, the degree of the equation is \(1\). Hence, the order and degree of the given differential equation are \(2\) and \(1\), respectively, and the correct answer is (C).