Question

\[5\frac{d^2 y}{d x^2} = \left\{ 1 + \left( \frac{dy}{dx} \right)^2 \right\}^{3/2}\]

Answer 1

\[5\frac{d^2 y}{d x^2} = \left\{ 1 + \left( \frac{dy}{dx} \right)^2 \right\}^\frac{3}{2} \]
Squaring both sides, we get
\[ \Rightarrow 25 \left( \frac{d^2 y}{d x^2} \right)^2 = \left\{ 1 + \left( \frac{dy}{dx} \right)^2 \right\}^3 \]
\[ \Rightarrow 25 \left( \frac{d^2 y}{d x^2} \right)^2 = 1 + 3 \left( \frac{dy}{dx} \right)^2 + 3 \left( \frac{dy}{dx} \right)^4 + \left( \frac{dy}{dx} \right)^6 \]
\[ \Rightarrow 25 \left( \frac{d^2 y}{d x^2} \right)^2 - \left( \frac{dy}{dx} \right)^6 - 3 \left( \frac{dy}{dx} \right)^4 - 3 \left( \frac{dy}{dx} \right)^2 - 1 = 0\]
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.
It is a non-linear differential equation, as its degree is 2, which is greater than 1.