Question

\[2\frac{d^2 y}{d x^2} + 3\sqrt{1 - \left( \frac{dy}{dx} \right)^2 - y} = 0\]

Answer 1

\[2\frac{d^2 y}{d x^2} + 3\sqrt{1 - \left( \frac{dy}{dx} \right)^2 - y} = 0\]
\[ \Rightarrow 2\frac{d^2 y}{d x^2} = - 3\sqrt{1 - \left( \frac{dy}{dx} \right)^2 - y}\]
Squaring both sides, we get
\[ \Rightarrow 4 \left( \frac{d^2 y}{d x^2} \right)^2 = 9\left[ 1 - \left( \frac{dy}{dx} \right)^2 - y \right]\]
\[ \Rightarrow 4 \left( \frac{d^2 y}{d x^2} \right)^2 + 9 \left( \frac{dy}{dx} \right)^2 + 9y - 9 = 0\]
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 it has degree 2, which is greater than 1.