The order and degree of the differential equation \( \left[2+\left(\displaystyle\frac{dy}{dx}\right)\right]^{5/2}=7\displaystyle\frac{d^{3}y}{dx^{3}} \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Order = 3, degree = 2
The highest derivative is \( \frac{d^{3}y}{dx^{3}} \), so the order is 3.
The degree is read only after the fractional power is removed. Squaring both sides,
\( \displaystyle \left[2+\frac{dy}{dx}\right]^{5}=49\left(\frac{d^{3}y}{dx^{3}}\right)^{2} \)
The highest derivative now appears to the power 2, so the degree is 2.
Why the other options are incorrect
Option (1) Order = 1, degree = 5 — Incorrect. This looks only at \( \frac{dy}{dx} \) and its power; the highest derivative is of third order.
Option (2) Order = 5, degree = 1 — Incorrect. The order is 3; 5 is the power of the bracket, not the order of a derivative.
Option (4) Order = 2, degree = 3 — Incorrect. This swaps the order (3) and the degree (2).