Determine the order and the degree of the equation $1-\left(\frac{dy}{dx}\right)^2=\left(a\frac{d^2y}{dx^2}\right)^{1/3}$.

Order = 2, Degree = 1

The correct answer is Option (2) → Order = 2, Degree = 1

The highest order derivative present in the given differential is $\frac{d^2y}{dx^2}$ so its order is 2. The given differential equation can be written as $\left(1-\left(\frac{dy}{dx}\right)^2\right)^3=a\frac{d^2y}{dx^2}$

or $1-\left(\frac{dy}{dx}\right)^6-3\left(\frac{dy}{dx}\right)^2\left(1-\left(\frac{dy}{dx}\right)^2\right)=a\frac{d^2y}{dx^2}$

or $a\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^6-3\left(\frac{dy}{dx}\right)^4+3\left(\frac{dy}{dx}\right)^2-1=0$.

Here each term in the derivative is a polynomial, so its degree is the highest exponent of $\frac{d^2y}{dx^2}$ which is 1. Thus, its degree is 1.

Hence, the order is 2 and degree is 1.