Match List-I with List-II
| List-I | List-II |
| Differential Equation | Order and degree |
| (A) \( \left(\displaystyle\frac{d^2y}{dx^2}\right)^2+e^{\frac{dy}{dx}}=0 \) | (I) order = 2 , degree = 1 |
| (B) \( xy\displaystyle\frac{d^2y}{dx^2}+x\left(\displaystyle\frac{dy}{dx}\right)^2-y\displaystyle\frac{dy}{dx}=0 \) | (II) order = 2 , degree = not defined |
| (C) \( \left(\displaystyle\frac{d^3y}{dx^3}\right)^2+\log_e\left(\displaystyle\frac{dy}{dx}\right)=4x \) | (III) order = 3 , degree = 2 |
| (D) \( \displaystyle\frac{d}{dx}\left(\displaystyle\frac{d^2y}{dx^2}\right)+y\displaystyle\frac{dy}{dx}=\left(1+\displaystyle\frac{dy}{dx}\right)^{1/2} \) | (IV) order = 3 , degree = not defined |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) - (II), (B) - (I), (C) - (IV), (D) - (III)
The degree is not defined when a derivative appears inside a function such as \( e^{u} \) or \( \log u \), because the equation is then not a polynomial in the derivatives.
(A) \( \left(\displaystyle\frac{d^2y}{dx^2}\right)^2+e^{\frac{dy}{dx}}=0 \)
This matches (II) order = 2, degree = not defined. The highest derivative is second order, but \( \frac{dy}{dx} \) appears in an exponent.
(B) \( xy\displaystyle\frac{d^2y}{dx^2}+x\left(\displaystyle\frac{dy}{dx}\right)^2-y\displaystyle\frac{dy}{dx}=0 \)
This matches (I) order = 2, degree = 1. The highest derivative \( \frac{d^2y}{dx^2} \) appears to the first power.
(C) \( \left(\displaystyle\frac{d^3y}{dx^3}\right)^2+\log_e\left(\displaystyle\frac{dy}{dx}\right)=4x \)
This matches (IV) order = 3, degree = not defined. The highest derivative is third order, but \( \frac{dy}{dx} \) appears inside a logarithm.
(D) \( \displaystyle\frac{d}{dx}\left(\displaystyle\frac{d^2y}{dx^2}\right)+y\displaystyle\frac{dy}{dx}=\left(1+\displaystyle\frac{dy}{dx}\right)^{1/2} \)
This matches (III) order = 3, degree = 2. The first term is \( \frac{d^3y}{dx^3} \). Squaring both sides removes the power 1/2:
\( \displaystyle \left(\frac{d^3y}{dx^3}+y\frac{dy}{dx}\right)^2=1+\frac{dy}{dx} \)
The third derivative now appears to the power 2.
So the matching is (A) - (II), (B) - (I), (C) - (IV), (D) - (III).