Question

\[\sqrt[3]{\frac{d^2 y}{d x^2}} = \sqrt{\frac{dy}{dx}}\]

Answer 1

\[\sqrt[3]{\frac{d^2 y}{d x^2}} = \sqrt{\frac{dy}{dx}}\]

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

Taking cubes of both the sides, we get

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

Squaring both the sides, we get

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

\[ \Rightarrow \left( \frac{d^2 y}{d x^2} \right)^2 - \left( \frac{dy}{dx} \right)^3 = 0\]

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.

Thus, it is a non-linear differential equation, as its degree is 2, which is greater than 1.