Match the LIST-I with LIST-II
| LIST-I (Differential Equation) | LIST-II (Order and degree) | ||
| A. | \( \log\left(\displaystyle\frac{dy}{dx}\right)+\displaystyle\frac{d^{2}y}{dx^{2}}=0 \) | I. | Order : 3 and degree : not defined |
| B. | \( \displaystyle\frac{d^{3}y}{dx^{3}}+y^{2}+e^{\frac{dy}{dx}}=4 \) | II. | Order : 2 and degree : 2 |
| C. | \( \displaystyle\frac{d}{dx}\left(\displaystyle\frac{dy}{dx}\right)^{2}+\displaystyle\frac{dy}{dx}=\sqrt{x} \) | III. | Order : 2 and degree : not defined |
| D. | \( \left(1+\left(\displaystyle\frac{dy}{dx}\right)^{2}\right)^{3/2}=\displaystyle\frac{d^{2}y}{dx^{2}} \) | IV. | Order : 2 and degree : 1 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A-III, B-I, C-IV, D-II
The order is the order of the highest derivative. The degree is the power of that derivative, and it is defined only when the equation is a polynomial in the derivatives.
(A) \( \log\left(\frac{dy}{dx}\right)+\frac{d^{2}y}{dx^{2}}=0 \)
This matches (III) Order : 2 and degree : not defined. The highest derivative is \( \frac{d^{2}y}{dx^{2}} \), and \( \log\left(\frac{dy}{dx}\right) \) is not a polynomial in \( \frac{dy}{dx} \).
(B) \( \frac{d^{3}y}{dx^{3}}+y^{2}+e^{\frac{dy}{dx}}=4 \)
This matches (I) Order : 3 and degree : not defined. The highest derivative is \( \frac{d^{3}y}{dx^{3}} \), and \( e^{\frac{dy}{dx}} \) is not a polynomial in \( \frac{dy}{dx} \).
(C) \( \frac{d}{dx}\left(\frac{dy}{dx}\right)^{2}+\frac{dy}{dx}=\sqrt{x} \)
This matches (IV) Order : 2 and degree : 1. Carrying out the differentiation first (chain rule):
\( \displaystyle \frac{d}{dx}\left(\frac{dy}{dx}\right)^{2}=2\,\frac{dy}{dx}\cdot\frac{d^{2}y}{dx^{2}} \)
\( \displaystyle 2\,\frac{dy}{dx}\cdot\frac{d^{2}y}{dx^{2}}+\frac{dy}{dx}=\sqrt{x} \)
The highest derivative \( \frac{d^{2}y}{dx^{2}} \) now appears to the power 1.
(D) \( \left(1+\left(\frac{dy}{dx}\right)^{2}\right)^{3/2}=\frac{d^{2}y}{dx^{2}} \)
This matches (II) Order : 2 and degree : 2. Squaring both sides removes the fractional power:
\( \displaystyle \left(1+\left(\frac{dy}{dx}\right)^{2}\right)^{3}=\left(\frac{d^{2}y}{dx^{2}}\right)^{2} \)
So the matching is (A)-(III), (B)-(I), (C)-(IV), (D)-(II).
Tip: In (C) the square is on \( \frac{dy}{dx} \), not on the highest derivative; expand the derivative first, then read the order and degree.