Match List-I with List-II
| List-I | List-II | ||
| Differential equation | Order and degree of differential equation | ||
| (A) | \( \displaystyle\frac{d^2y}{dx^2}+4y=0 \) | (I) | Order = 2 degree = 3 |
| (B) | \( \displaystyle\frac{d^3y}{dx^3}+\left(\frac{dy}{dx}\right)^2+xy=0 \) | (II) | Order = 2 degree = 2 |
| (C) | \( \left(\displaystyle\frac{d^2y}{dx^2}\right)^{\frac{1}{2}}=\left(\displaystyle\frac{dy}{dx}\right)^{\frac{1}{3}} \) | (III) | Order = 2 degree = 1 |
| (D) | \( \sqrt{1+\left(\displaystyle\frac{dy}{dx}\right)^2}=\left(k\,\displaystyle\frac{d^2y}{dx^2}\right)^{\frac{1}{3}} \) | (IV) | Order = 3 degree = 1 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) - (III), (B) - (IV), (C) - (I), (D) - (II)
The order is the order of the highest derivative. The degree is the power of that highest derivative after the equation is made free of fractional powers of the derivatives.
(A) \( \frac{d^2y}{dx^2}+4y=0 \)
This matches (III) Order = 2 degree = 1.
The highest derivative \( \frac{d^2y}{dx^2} \) appears to power 1.
(B) \( \frac{d^3y}{dx^3}+\left(\frac{dy}{dx}\right)^2+xy=0 \)
This matches (IV) Order = 3 degree = 1.
The highest derivative is \( \frac{d^3y}{dx^3} \), to power 1. The square on \( \frac{dy}{dx} \) does not decide the degree.
(C) \( \left(\frac{d^2y}{dx^2}\right)^{\frac{1}{2}}=\left(\frac{dy}{dx}\right)^{\frac{1}{3}} \)
This matches (I) Order = 2 degree = 3.
Raising both sides to the power 6 (the LCM of 2 and 3):
\( \displaystyle \left(\frac{d^2y}{dx^2}\right)^{3}=\left(\frac{dy}{dx}\right)^{2} \)
(D) \( \sqrt{1+\left(\frac{dy}{dx}\right)^2}=\left(k\,\frac{d^2y}{dx^2}\right)^{\frac{1}{3}} \)
This matches (II) Order = 2 degree = 2.
Raising both sides to the power 6:
\( \displaystyle \left[1+\left(\frac{dy}{dx}\right)^2\right]^{3}=k^2\left(\frac{d^2y}{dx^2}\right)^{2} \)
So the matching is (A)-(III), (B)-(IV), (C)-(I), (D)-(II).
Tip: Never read the degree off an equation that still has fractional powers of derivatives: clear them first, as in (C) and (D), where the printed powers 1/2 and 1/3 are not the degree.