The order and the degree of the differential equation \(\sqrt{1 + \left(\displaystyle\frac{dy}{dx}\right)^3} = \displaystyle\frac{d^2y}{dx^2}\) are
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → order 2, degree 2
The highest derivative present is \(\frac{d^2y}{dx^2}\), so the order is 2.
Before reading the degree, the equation must be made free from the radical. Squaring both sides:
\( \displaystyle 1 + \left(\frac{dy}{dx}\right)^3 = \left(\frac{d^2y}{dx^2}\right)^2 \)
This is a polynomial in the derivatives, and the highest derivative \(\frac{d^2y}{dx^2}\) has power 2. So the degree is 2.
Why the other options are incorrect
Option (1) order 2, degree 1 — Incorrect. The power 1 of \(\frac{d^2y}{dx^2}\) is read before the square root is removed; after squaring the power is 2.
Option (2) order 1, degree 3 — Incorrect. This uses the first derivative and its power 3, but the second derivative is the highest one present.
Option (3) order 2, degree not defined — Incorrect. A square root of a polynomial expression is removed by squaring, after which the equation is a polynomial in the derivatives, so the degree is defined.