Match List-I with List-II
| List-I | List-II | ||
| Differential equation | The sum of degree and order of the differential equation | ||
| (A) | \( x\displaystyle\frac{d^2y}{dx^2}=\left(1+\left(\displaystyle\frac{dy}{dx}\right)^2\right)^{\frac{1}{3}} \) | (I) | 2 |
| (B) | \( x\displaystyle\frac{d^2y}{dx^2}=\left(2-\displaystyle\frac{dy}{dx}\right)^{\frac{1}{7}} \) | (II) | 9 |
| (C) | \( 3+\displaystyle\frac{dy}{dx}=y+\displaystyle\frac{d^3y}{dx^3} \) | (III) | 4 |
| (D) | \( \displaystyle\frac{dy}{dx}=x+3 \) | (IV) | 5 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) - (IV), (B) - (II), (C) - (III), (D) - (I)
The degree can be read only when the equation is free of fractional powers of the derivatives, so those powers are removed first.
(A) \( x\frac{d^2y}{dx^2}=\left(1+\left(\frac{dy}{dx}\right)^2\right)^{\frac{1}{3}} \)
This matches (IV) 5. Cubing both sides:
\( \displaystyle x^3\left(\frac{d^2y}{dx^2}\right)^3=1+\left(\frac{dy}{dx}\right)^2 \)
Order 2, degree 3, so the sum is 5.
(B) \( x\frac{d^2y}{dx^2}=\left(2-\frac{dy}{dx}\right)^{\frac{1}{7}} \)
This matches (II) 9. Raising both sides to the power 7:
\( \displaystyle x^7\left(\frac{d^2y}{dx^2}\right)^7=2-\frac{dy}{dx} \)
Order 2, degree 7, so the sum is 9.
(C) \( 3+\frac{dy}{dx}=y+\frac{d^3y}{dx^3} \)
This matches (III) 4. The highest derivative is \( \frac{d^3y}{dx^3} \), to the power 1: order 3, degree 1, sum 4.
(D) \( \frac{dy}{dx}=x+3 \)
This matches (I) 2. Order 1, degree 1, sum 2.
So the matching is (A)-(IV), (B)-(II), (C)-(III), (D)-(I).