Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | The degree of the differential equation \(\left(\displaystyle\frac{d^2y}{dx^2}\right)^2 + \left(\displaystyle\frac{dy}{dx}\right)^2 = x\sin\left(\displaystyle\frac{dy}{dx}\right)\) | (I) | 4 |
| (B) | The degree of differential equation \(\displaystyle\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^{1/4} + x^{1/5} = 0\) | (II) | 1 |
| (C) | The degree of differential equation \(\displaystyle\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 + 6y^5 = 0\) | (III) | Not defined |
| (D) | The degree of differential equation \(1 + \left(\displaystyle\frac{dy}{dx}\right)^4 = 7\left(\displaystyle\frac{d^2y}{dx^2}\right)^3\) | (IV) | 3 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(III), (B)-(I), (C)-(II), (D)-(IV)
The degree of a differential equation is the highest power of the highest order derivative, once the equation is a polynomial in the derivatives (fractional powers of derivatives removed). If it cannot be made a polynomial in the derivatives, the degree is not defined. In all four equations the highest order derivative is \( \frac{d^2y}{dx^2} \).
(A) \(\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = x\sin\left(\frac{dy}{dx}\right)\)
This matches (III) Not defined. The term \(\sin\left(\frac{dy}{dx}\right)\) is not a polynomial in \( \frac{dy}{dx} \).
(B) \(\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^{1/4} + x^{1/5} = 0\)
This matches (I) 4. Isolating the fractional power of the derivative and raising both sides to the power 4:
\( \displaystyle \left(\frac{dy}{dx}\right)^{1/4} = -\left(\frac{d^2y}{dx^2} + x^{1/5}\right) \)
\( \displaystyle \frac{dy}{dx} = \left(\frac{d^2y}{dx^2} + x^{1/5}\right)^4 \)
Now \( \frac{d^2y}{dx^2} \) appears with highest power 4. (The power \(x^{1/5}\) is a power of x, not of a derivative, so it does not matter.)
(C) \(\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 + 6y^5 = 0\)
This matches (II) 1. The equation is already a polynomial in the derivatives, and \( \frac{d^2y}{dx^2} \) appears to the power 1.
(D) \(1 + \left(\frac{dy}{dx}\right)^4 = 7\left(\frac{d^2y}{dx^2}\right)^3\)
This matches (IV) 3. \( \frac{d^2y}{dx^2} \) appears to the power 3; the power 4 belongs to the lower order derivative \( \frac{dy}{dx} \).
So the matching is (A)-(III), (B)-(I), (C)-(II), (D)-(IV).