Question

$y=x\frac{dy}{dx}+a\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}$

Answer 1

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

In this differential equation, the order of the highest order derivative is 1 and its highest power is 2. So, it is a differential equation of order 1 and degree 2.

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