The order and degree of the differential equation \( \displaystyle\frac{d^3y}{dx^3}+\left(\frac{d^2y}{dx^2}\right)^2+e^{\frac{dy}{dx}}=2 \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Order = 3 ; degree is not defined
The highest derivative present is \( \frac{d^3y}{dx^3} \), so the order is 3.
The degree is defined only when the equation is a polynomial in its derivatives. The term \( e^{\frac{dy}{dx}} \) is an exponential function of the derivative \( \frac{dy}{dx} \) and cannot be written as a polynomial in it. So the degree is not defined.
Why the other options are incorrect
Option (1) Order = 2 ; degree = 2 — Incorrect. The highest derivative is the third derivative, not the second, so the order is 3; the power 2 on \( \frac{d^2y}{dx^2} \) does not give the degree.
Option (2) Order = 3 ; degree = 1 — Incorrect. The order is correct, but the term \( e^{\frac{dy}{dx}} \) is not a polynomial in \( \frac{dy}{dx} \), so the degree cannot be 1.
Option (4) Order = 1 ; degree = 3 — Incorrect. The order and degree are interchanged, and the degree is not defined in any case.