Match the List-I with List-II
| List-I | List-II | ||
| Differential Equation | Degree of Differential Equation | ||
| (A) | \( \displaystyle\frac{d^{3}y}{dx^{3}}+\left(\displaystyle\frac{d^{2}y}{dx^{2}}\right)^{2}-\displaystyle\frac{dy}{dx}+y=0 \) | (I) | 2 |
| (B) | \( \displaystyle\frac{dy}{dx}+\left(\displaystyle\frac{d^{2}y}{dx^{2}}\right)^{4}+2\left(\displaystyle\frac{d^{3}y}{dx^{3}}\right)^{3}=0 \) | (II) | 4 |
| (C) | \( \left(\displaystyle\frac{d^{3}y}{dx^{3}}\right)^{2}+\left(\displaystyle\frac{d^{2}y}{dx^{2}}\right)^{4}-y=0 \) | (III) | 1 |
| (D) | \( \left(\displaystyle\frac{d^{2}y}{dx^{2}}\right)^{4}+\left(\displaystyle\frac{dy}{dx}\right)^{3}+y=0 \) | (IV) | 3 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) - (III), (B) - (IV), (C) - (I), (D) - (II)
All four equations are polynomial in their derivatives. The degree is the power of the highest order derivative, not the largest power appearing anywhere.
(A) \( \frac{d^{3}y}{dx^{3}}+\left(\frac{d^{2}y}{dx^{2}}\right)^{2}-\frac{dy}{dx}+y=0 \)
This matches (III) 1.
The highest order derivative is \( \frac{d^{3}y}{dx^{3}} \) (order 3), and it appears to the power 1.
(B) \( \frac{dy}{dx}+\left(\frac{d^{2}y}{dx^{2}}\right)^{4}+2\left(\frac{d^{3}y}{dx^{3}}\right)^{3}=0 \)
This matches (IV) 3.
The highest order derivative is \( \frac{d^{3}y}{dx^{3}} \), and it appears to the power 3. The power 4 belongs to the lower-order \( \frac{d^{2}y}{dx^{2}} \).
(C) \( \left(\frac{d^{3}y}{dx^{3}}\right)^{2}+\left(\frac{d^{2}y}{dx^{2}}\right)^{4}-y=0 \)
This matches (I) 2.
The highest order derivative \( \frac{d^{3}y}{dx^{3}} \) appears to the power 2.
(D) \( \left(\frac{d^{2}y}{dx^{2}}\right)^{4}+\left(\frac{dy}{dx}\right)^{3}+y=0 \)
This matches (II) 4.
The highest order derivative is \( \frac{d^{2}y}{dx^{2}} \) (order 2), and it appears to the power 4.
So the matching is (A) - (III), (B) - (IV), (C) - (I), (D) - (II).