Question

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

Answer 1

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

\[ \Rightarrow x + \left( \frac{dy}{dx} \right) = \left( 1 + \left( \frac{dy}{dx} \right)^2 \right)^\frac{1}{2} \]

Squaring both sides, we get

\[ \Rightarrow \left( x + \frac{dy}{dx} \right)^2 = 1 + \left( \frac{dy}{dx} \right)^2 \]

\[ \Rightarrow x^2 + 2x\frac{dy}{dx} + \left( \frac{dy}{dx} \right)^2 = 1 + \left( \frac{dy}{dx} \right)^2 \]

\[ \Rightarrow 2x\frac{dy}{dx} + x^2 = 1\]

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.