The sum of order and degree of the differential equation \(y = x\displaystyle\frac{dy}{dx} + 2\sqrt{1 + \left(\frac{dy}{dx}\right)^2}\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 3
Let \(p = \frac{dy}{dx}\). The only derivative is \(\frac{dy}{dx}\), so the order is 1.
The degree is found only after the equation is made free of radicals. Isolating the root and squaring:
\(y - xp = 2\sqrt{1 + p^2}\)
\((y - xp)^2 = 4(1 + p^2)\)
\((x^2 - 4)p^2 - 2xyp + y^2 - 4 = 0\)
The highest power of p is 2, so the degree is 2.
Order + degree = 1 + 2 = 3
Tip: Read the degree only after removing the square root. Reading it from the printed form gives degree 1 and the wrong sum 2 (Option (2)).