The differential equation whose order and degree are 3 and 2 respectively is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{d}{dx}\left(\displaystyle\frac{d^2y}{dx^2}\right)+y\displaystyle\frac{dy}{dx}=\left(1+\displaystyle\frac{dy}{dx}\right)^{1/2} \)
The order is the highest derivative present; the degree is the power of that highest derivative once the equation is a polynomial in the derivatives (no roots, fractional powers, log, exponential of a derivative).
In Option (3), \( \frac{d}{dx}\left(\frac{d^2y}{dx^2}\right)=\frac{d^3y}{dx^3} \), so the equation is
\( \displaystyle \frac{d^3y}{dx^3}+y\frac{dy}{dx}=\left(1+\frac{dy}{dx}\right)^{1/2} \)
Squaring both sides removes the fractional power:
\( \displaystyle \left(\frac{d^3y}{dx^3}+y\frac{dy}{dx}\right)^2=1+\frac{dy}{dx} \)
The highest derivative is the third one and its highest power is 2. So the order is 3 and the degree is 2.
Why the other options are incorrect
Option (1) \( \left(\displaystyle\frac{d^2y}{dx^2}\right)^2+e^{\frac{dy}{dx}}=y \) — Incorrect. The highest derivative is \( \frac{d^2y}{dx^2} \), so the order is 2; and because of \( e^{dy/dx} \) the degree is not defined.
Option (2) \( \left(\displaystyle\frac{d^3y}{dx^3}\right)^2+\log\left(\displaystyle\frac{dy}{dx}\right)=4x \) — Incorrect. The order is 3, but \( \log\left(\frac{dy}{dx}\right) \) is not a polynomial in the derivatives, so the degree is not defined.
Option (4) \( \displaystyle\frac{d^3y}{dx^3}+x\displaystyle\frac{d^2y}{dx^2}+\left(\displaystyle\frac{dy}{dx}\right)^2=\log x \) — Incorrect. The order is 3, but \( \frac{d^3y}{dx^3} \) appears only to the power 1, so the degree is 1 (the square on \( \frac{dy}{dx} \) does not count, since it is not the highest derivative).