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Question
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Solution
\[x + \left( \frac{dy}{dx} \right) = \sqrt{1 + \left( \frac{dy}{dx} \right)^2}\]
\[ \Rightarrow x + \left( \frac{dy}{dx} \right) = \left( 1 + \left( \frac{dy}{dx} \right)^2 \right)^\frac{1}{2} \]
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.
Notes
The answer given in the book has some error. The solution here is created according to the question given in the book.
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