Question

The degree of the differential equation \(\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = x\sin\left(\frac{d^2y}{dx^2}\right)\) is

• A. 1
• B. 2
• C. 3
• D. None of these

Hint:

Degree is defined only when the differential equation is a polynomial in derivatives.

Answer 1

The Correct Option is D

The degree of a differential equation is defined as the highest power of the highest order derivative present in the equation after it has been made free from radicals and fractions in relation to the derivatives.

Consider the given differential equation: \(\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = x\sin\left(\frac{d^2y}{dx^2}\right)\)

To determine the degree, we first need to ensure the equation is free from any non-polynomial terms. In this equation, the sine function \(\sin\left(\frac{d^2y}{dx^2}\right)\) makes it non-polynomial with respect to the highest order derivative, which is \(\frac{d^2y}{dx^2}\).

Since the equation cannot be expressed in polynomial form with respect to the highest and other derivatives, the degree of this differential equation is not defined. As a result, the correct answer is:

None of these

This is because the presence of the sine function alongside a second derivative makes it impossible to express the equation as a polynomial of just \(\frac{d^2y}{dx^2}\) and \(\frac{dy}{dx}\) terms. Hence, this results in the degree being undefined.