Match List-I with List-II
| List-I | List-II |
| Differential equation | Order and Degree |
| (A) \( \displaystyle\frac{d^2y}{dx^2}=1+\sqrt{\displaystyle\frac{dy}{dx}} \) | (I) Order = 2, Degree = 1 |
| (B) \( \displaystyle\frac{dy}{dx}+2\displaystyle\frac{dx}{dy}=x \) | (II) Order = 1, degree = 1 |
| (C) \( y+2\displaystyle\frac{dy}{dx}=\int y\,dx \) | (III) Order = 1, degree = 2 |
| (D) \( \displaystyle\frac{dy}{dx}+y=\log x \) | (IV) Order = 2, degree = 2 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) - (IV), (B) - (III), (C) - (I), (D) - (II)
The degree is defined only when the equation is a polynomial in the derivatives, so roots, reciprocals and integrals are removed first.
(A) \( \displaystyle\frac{d^2y}{dx^2}=1+\sqrt{\displaystyle\frac{dy}{dx}} \)
This matches (IV) Order = 2, degree = 2. Remove the root by squaring:
\( \displaystyle \left(\frac{d^2y}{dx^2}-1\right)^2=\frac{dy}{dx} \)
(B) \( \displaystyle\frac{dy}{dx}+2\displaystyle\frac{dx}{dy}=x \)
This matches (III) Order = 1, degree = 2. Since \( \frac{dx}{dy}=\frac{1}{dy/dx} \), multiply by \( \frac{dy}{dx} \):
\( \displaystyle \left(\frac{dy}{dx}\right)^2+2=x\frac{dy}{dx} \)
(C) \( y+2\displaystyle\frac{dy}{dx}=\int y\,dx \)
This matches (I) Order = 2, degree = 1. Differentiate once to remove the integral:
\( \displaystyle \frac{dy}{dx}+2\frac{d^2y}{dx^2}=y \)
(D) \( \displaystyle\frac{dy}{dx}+y=\log x \)
This matches (II) Order = 1, degree = 1. Only the first derivative appears, to power 1.
So the matching is (A) - (IV), (B) - (III), (C) - (I), (D) - (II).