Match List-I with List-II
| List-I | List-II | ||
| Differential Equation | Order and Degree | ||
| (A) | \(\left(\displaystyle\frac{d^2y}{dx^2}\right)^2-e^x\left(\displaystyle\frac{dy}{dx}\right)^4+1=0\) | (I) | order = 1 and degree = 2 |
| (B) | \(\left(\displaystyle\frac{dy}{dx}\right)^2+xy = 0\) | (II) | order = 2 and degree = 1 |
| (C) | \(\left(1+\displaystyle\frac{dy}{dx}\right)^{3/2}=4\left(\displaystyle\frac{d^2y}{dx^2}\right)^2\) | (III) | order = 2 and degree = 2 |
| (D) | \(\sqrt{\displaystyle\frac{d^2y}{dx^2}+1}=\displaystyle\frac{dy}{dx}\) | (IV) | order = 2 and degree = 4 |
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 highest derivative once the equation is a polynomial in its derivatives (no fractional powers or roots of derivatives).
(A) \(\left(\frac{d^2y}{dx^2}\right)^2-e^x\left(\frac{dy}{dx}\right)^4+1=0\)
This matches (III) order = 2 and degree = 2. The highest derivative is \(\frac{d^2y}{dx^2}\), and its power is 2. (The power 4 belongs to \(\frac{dy}{dx}\), which is not the highest derivative.)
(B) \(\left(\frac{dy}{dx}\right)^2+xy = 0\)
This matches (I) order = 1 and degree = 2. The only derivative is \(\frac{dy}{dx}\), and its power is 2.
(C) \(\left(1+\frac{dy}{dx}\right)^{3/2}=4\left(\frac{d^2y}{dx^2}\right)^2\)
This matches (IV) order = 2 and degree = 4. Squaring both sides removes the power 3/2:
\( \displaystyle \left(1+\frac{dy}{dx}\right)^{3}=16\left(\frac{d^2y}{dx^2}\right)^4 \)
The highest derivative \(\frac{d^2y}{dx^2}\) now has power 4.
(D) \(\sqrt{\frac{d^2y}{dx^2}+1}=\frac{dy}{dx}\)
This matches (II) order = 2 and degree = 1. Squaring both sides removes the root:
\( \displaystyle \frac{d^2y}{dx^2}+1=\left(\frac{dy}{dx}\right)^2 \)
The highest derivative \(\frac{d^2y}{dx^2}\) now has power 1.
So the matching is (A) - (III), (B) - (I), (C) - (IV), (D) - (II).