The order of the differential equation \( \left(\displaystyle\frac{d^3y}{dx^3}\right)^4+\displaystyle\frac{d^4y}{dx^4}+\log\left(\displaystyle\frac{dy}{dx}\right)=0 \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 4
The order of a differential equation is the order of the highest order derivative present in it.
The derivatives present are \( \frac{d^3y}{dx^3} \) (order 3), \( \frac{d^4y}{dx^4} \) (order 4) and \( \frac{dy}{dx} \) (order 1).
The highest of these is \( \frac{d^4y}{dx^4} \), so the order is 4.
Why the other options are incorrect
Option (2) 3 — Incorrect. 3 is the order of \( \frac{d^3y}{dx^3} \), but a higher derivative \( \frac{d^4y}{dx^4} \) is also present; the power 4 on the first term does not change its order.
Option (3) 1 — Incorrect. 1 is the order of \( \frac{dy}{dx} \) inside the logarithm, the lowest derivative, not the highest.
Option (4) Not defined — Incorrect. It is the degree that is not defined here, because \( \frac{dy}{dx} \) sits inside a logarithm; the order is always defined.