Which of the following differential equations is of order 3 and degree 1?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{d^3y}{dx^3} + \left(\frac{d^2y}{dx^2}\right)^2 = x\)
The order is the order of the highest derivative, and the degree is the power of that highest derivative.
In Option (1) the highest derivative is \(\frac{d^3y}{dx^3}\), so the order is 3.
It appears with power 1, so the degree is 1. The square is on \(\frac{d^2y}{dx^2}\), a lower derivative.
Why the other options are incorrect
Option (2) \(\left(\frac{d^2y}{dx^2}\right)^3 + \frac{dy}{dx} = 0\) — Incorrect. The highest derivative is the second, so the order is 2 (and the degree is 3).
Option (3) \(\left(\frac{d^3y}{dx^3}\right)^2 + y = 0\) — Incorrect. The order is 3, but \(\frac{d^3y}{dx^3}\) is squared, so the degree is 2.
Option (4) \(\left(\frac{dy}{dx}\right)^3 = x + y\) — Incorrect. Only the first derivative is present, so the order is 1 (and the degree is 3).