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Question
Write the order and degree of the differential equation
\[y = x\frac{dy}{dx} + a\sqrt{1 + \left( \frac{dy}{dx} \right)^2}\]
Solution
We have,
\[y = x\frac{dy}{dx} + a\sqrt{1 + \left( \frac{dy}{dx} \right)^2}\]
\[y - x\frac{dy}{dx} = a\sqrt{1 + \left( \frac{dy}{dx} \right)^2}\]
Squaring both sides, we get
\[ \left( y - x\frac{dy}{dx} \right)^2 = \left\{ a\sqrt{1 + \left( \frac{dy}{dx} \right)^2} \right\}^2 \]
\[ y^2 + x^2 \left( \frac{dy}{dx} \right)^2 - 2xy\frac{dy}{dx} = a^2 \left\{ 1 + \left( \frac{dy}{dx} \right)^2 \right\}\]
\[ y^2 + \left( \frac{dy}{dx} \right)^2 \left\{ x^2 - a^2 \right\} - 2xy\frac{dy}{dx} = a^2 \]
From the above equation, we see that the highest order is 1 .
So, its order is 1 and the power of the highest order derivative is 2 .
Thus, it is a differential equation of order 1 and degree 2 .
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