Match List-I with List-II
| List-I | List-II | ||
| (A) | \( y+\displaystyle\frac{dy}{dx}=\displaystyle\frac{1}{4}\int y\,dx \) | (I) | Order 1, Degree 1 |
| (B) | \( y=\displaystyle\frac{dy}{dx}+\displaystyle\frac{C}{dy/dx} \) | (II) | Order 2, Degree 2 |
| (C) | \( (xy^2+x)dx+(y-x^2)dy=0 \) | (III) | Order 2, Degree 1 |
| (D) | \( \left[\left\{1+\left(\displaystyle\frac{dy}{dx}\right)^2\right\}^{3/2}\Big/\displaystyle\frac{d^2y}{dx^2}\right]=K \) | (IV) | Order 1, Degree 2 |
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, read only after the equation is free of integrals, fractions and fractional powers of the derivatives.
(A) \( y+\frac{dy}{dx}=\frac{1}{4}\int y\,dx \)
This matches (III) Order 2, Degree 1. The integral must be removed first, so we differentiate both sides with respect to x:
\( \displaystyle \frac{dy}{dx}+\frac{d^2y}{dx^2}=\frac{1}{4}y \)
The highest derivative is \( \frac{d^2y}{dx^2} \), to the power 1.
(B) \( y=\frac{dy}{dx}+\frac{C}{dy/dx} \)
This matches (IV) Order 1, Degree 2. Multiplying through by \( \frac{dy}{dx} \) clears the fraction:
\( \displaystyle y\frac{dy}{dx}=\left(\frac{dy}{dx}\right)^2+C \)
The only derivative is \( \frac{dy}{dx} \), and its highest power is 2.
(C) \( (xy^2+x)dx+(y-x^2)dy=0 \)
This matches (I) Order 1, Degree 1. Dividing by dx:
\( \displaystyle (xy^2+x)+(y-x^2)\frac{dy}{dx}=0 \)
Only \( \frac{dy}{dx} \) appears, to the power 1.
(D) \( \left[\left\{1+\left(\frac{dy}{dx}\right)^2\right\}^{3/2}\Big/\frac{d^2y}{dx^2}\right]=K \)
This matches (II) Order 2, Degree 2. Clearing the fraction, and then the power 3/2 by squaring:
\( \displaystyle \left\{1+\left(\frac{dy}{dx}\right)^2\right\}^{3/2}=K\frac{d^2y}{dx^2} \)
\( \displaystyle \left\{1+\left(\frac{dy}{dx}\right)^2\right\}^{3}=K^2\left(\frac{d^2y}{dx^2}\right)^2 \)
The highest derivative \( \frac{d^2y}{dx^2} \) now appears to the power 2.
So the matching is (A)-(III), (B)-(IV), (C)-(I), (D)-(II).