The sum of the order and degree of the differential equation : \( \left(x+\displaystyle\frac{dy}{dx}\right)^2=\displaystyle\frac{dy}{dx}+1 \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 3
The only derivative present is \( \frac{dy}{dx} \), so the order is 1.
Expanding the left side shows the powers of the derivative clearly:
\( \displaystyle x^2+2x\frac{dy}{dx}+\left(\frac{dy}{dx}\right)^2=\frac{dy}{dx}+1 \)
The equation is a polynomial in \( \frac{dy}{dx} \), and the highest power is 2, so the degree is 2.
\( \displaystyle \text{order}+\text{degree}=1+2=3 \)
Why the other options are incorrect
Option (1) 1 — Incorrect. The order and the degree are each at least 1, so their sum cannot be 1.
Option (3) 2 — Incorrect. This takes the degree as 1; after expanding the square, \( \left(\frac{dy}{dx}\right)^2 \) appears, so the degree is 2.
Option (4) 4 — Incorrect. This takes the order as 2, but the equation has no second derivative.