Question

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

Answer 1

$$\begin{gathered} 2\frac{d^{2}y}{d{x}^2}+3\sqrt{1-{\left(\frac{dy}{dx}\right)}^2-y}=0 \\ \Rightarrow 2\frac{d^{2}y}{d{x}^2}=-3\sqrt{1-{\left(\frac{dy}{dx}\right)}^2-y} \\ \mathrm{Squaring}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \Rightarrow 4{\left(\frac{d^{2}y}{d{x}^2}\right)}^2=9\left[1-{\left(\frac{dy}{dx}\right)}^2-y\right] \\ \Rightarrow 4{\left(\frac{d^{2}y}{d{x}^2}\right)}^2+9{\left(\frac{dy}{dx}\right)}^2+9y-9=0 \end{gathered}$$

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.