The degree of the differential equation \(\left(1-\left(\displaystyle\frac{dy}{dx}\right)^2\right)^{3/2}=k\displaystyle\frac{d^2y}{dx^2}\) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 2
The degree of a differential equation is the power of its highest order derivative, after the equation has been made free of fractional powers of the derivatives.
Here the left side has the power 3/2, so squaring both sides gives
\( \displaystyle \left(1-\left(\frac{dy}{dx}\right)^2\right)^{3} = k^2\left(\frac{d^2y}{dx^2}\right)^2 \)
The highest order derivative is \(\frac{d^2y}{dx^2}\), and its power is now 2. So the degree is 2.
Why the other options are incorrect
Option (1) 1 — Incorrect. The power of \(\frac{d^2y}{dx^2}\) is 1 only before the fractional power 3/2 is removed.
Option (3) 3 — Incorrect. 3 is the power of the bracket containing \(\frac{dy}{dx}\), which is not the highest order derivative.
Option (4) \(\displaystyle\frac{3}{2}\) — Incorrect. The degree is always a positive integer; 3/2 is the fractional power that has to be removed first.