Match the LIST-I with LIST-II
| LIST-I (Differential equation) | LIST-II (Degree) | ||
| A. | \( \left(\displaystyle\frac{dy}{dx}\right)^{2}+\displaystyle\frac{1}{\displaystyle\frac{dy}{dx}}=5 \) | I. | 2 |
| B. | \( \displaystyle\frac{d^{3}y}{dx^{3}}=\left\{3+\left(\displaystyle\frac{dy}{dx}\right)^{2}\right\}^{3/2} \) | II. | 4 |
| C. | \( \displaystyle\frac{d^{2}y}{dx^{2}}=\left(\displaystyle\frac{dy}{dx}\right)^{5/4} \) | III. | 1 |
| D. | \( y\displaystyle\frac{d^{2}x}{dy^{2}}=y^{2}+1 \) | IV. | 3 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → A-IV, B-I, C-II, D-III
The degree is read only after the equation is made a polynomial in its derivatives (no derivative in a denominator, no fractional powers).
(A) \( \left(\frac{dy}{dx}\right)^{2}+\frac{1}{\frac{dy}{dx}}=5 \)
This matches (IV) 3. Multiplying by \( \frac{dy}{dx} \):
\( \displaystyle \left(\frac{dy}{dx}\right)^{3}+1=5\frac{dy}{dx} \)
(B) \( \frac{d^{3}y}{dx^{3}}=\left\{3+\left(\frac{dy}{dx}\right)^{2}\right\}^{3/2} \)
This matches (I) 2. Squaring both sides:
\( \displaystyle \left(\frac{d^{3}y}{dx^{3}}\right)^{2}=\left\{3+\left(\frac{dy}{dx}\right)^{2}\right\}^{3} \)
(C) \( \frac{d^{2}y}{dx^{2}}=\left(\frac{dy}{dx}\right)^{5/4} \)
This matches (II) 4. Raising both sides to the power 4:
\( \displaystyle \left(\frac{d^{2}y}{dx^{2}}\right)^{4}=\left(\frac{dy}{dx}\right)^{5} \)
The power of the highest order derivative \( \frac{d^{2}y}{dx^{2}} \) is 4; the power 5 belongs to \( \frac{dy}{dx} \).
(D) \( y\frac{d^{2}x}{dy^{2}}=y^{2}+1 \)
This matches (III) 1. Here x is the dependent variable; the highest derivative \( \frac{d^{2}x}{dy^{2}} \) appears to the power 1.
So the matching is (A)-(IV), (B)-(I), (C)-(II), (D)-(III).