Question

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

Answer 1

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

Squaring both sides, we get

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

Taking cubes of both sides, we get

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

\[ \Rightarrow c^2 \left( \frac{d^2 y}{d x^2} \right)^2 = 1 + 3 \left( \frac{dy}{dx} \right)^2 + 3 \left( \frac{dy}{dx}\right)^4 + \left( \frac{dy}{dx} \right)^6\]

In this differential equation, the order of the highest order derivative is 2 and its power is 2. So, it is a differential equation of order 2 and degree 2.

It is a non-linear differential equation, as its degree is more than 1.