Question

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

Answer 1

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

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

Hence, it is a linear differential equation.

Disclaimer: The answer given in the book has some error. The solution here is created according to the question given in the book.