Question

The degree of the differential equation
\(1+(\frac{dy}{dx})^2+(\frac{d^2y}{dx^2})^2= \sqrt[3]{\frac{d^2y}{dx^2}+1}\) is

• A. 1
• B. 6
• C. 2
• D. 3

Hint:

To find the degree of a differential equation, first make sure all derivatives are polynomial, and eliminate any fractional powers or roots. The degree is determined by the highest power of the highest-order derivative in the equation.

Answer 1

The Correct Option is B

The correct answer is: (B): 6.

We are tasked with determining the degree of the differential equation:

\(1 + \left(\frac{dy}{dx}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^2 = \sqrt[3]{\frac{d^2y}{dx^2} + 1}\)

Step 1: Express the equation in terms of derivatives

To find the degree, we need to ensure that the equation is polynomial in the derivatives of \( y \). Start by clearing the cube root by raising both sides of the equation to the power of 3:

\( \left( 1 + \left(\frac{dy}{dx}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^2 \right)^3 = \frac{d^2y}{dx^2} + 1 \)

Step 2: Eliminate the cube root and simplify

By simplifying both sides of the equation, we can rewrite the equation in a form where the derivatives are only raised to integer powers. This step essentially removes the fractional powers and simplifies the equation to a polynomial in \( \frac{dy}{dx} \) and \( \frac{d^2y}{dx^2} \). The left side of the equation becomes a polynomial, while the right side has only the second derivative of \( y \), ensuring that we have a polynomial equation in terms of the derivatives.

Step 3: Identify the degree of the equation

The degree of a differential equation is defined as the highest power of the highest order derivative in the equation. In this case, after simplifying the equation, the highest order derivative is \( \frac{d^2y}{dx^2} \), and the highest power of this derivative is 6, making the degree of the equation 6.

Conclusion:
The degree of the differential equation is (B): 6.