Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Differential equation | Degree | ||
| (A) | \(\displaystyle\frac{d^2y}{dx^2} +\sqrt{\frac{dy}{dx}}-y=0\) | (I) | 6 |
| (B) | \(\displaystyle\sqrt{\frac{d^3y}{dx^3}}-\sqrt[12]{\frac{d^2y}{dx^2}}=0\) | (II) | Not defined |
| (C) | \(\displaystyle\left(\frac{d^2y}{dx^2}\right)^2+\frac{dy}{dx}+e^{\frac{dy}{dx}}=x^2\) | (III) | 3 |
| (D) | \(\displaystyle\sqrt[3]{\frac{dy}{dx}}-\frac{d^2y}{dx^2}=e^x\) | (IV) | 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)-(I), (C)-(II), (D)-(III)
The degree is the power of the highest order derivative once the equation is a polynomial in the derivatives (radicals removed). It is not defined if the equation cannot be made a polynomial in the derivatives.
(A) \(\displaystyle\frac{d^2y}{dx^2} +\sqrt{\frac{dy}{dx}}-y=0\)
This matches (IV) 2. Isolating the radical and squaring:
\( \displaystyle \sqrt{\frac{dy}{dx}} = y - \frac{d^2y}{dx^2} \)
\( \displaystyle \frac{dy}{dx} = \left(y - \frac{d^2y}{dx^2}\right)^2 \)
The highest order derivative \(\frac{d^2y}{dx^2}\) now appears with power 2.
(B) \(\displaystyle\sqrt{\frac{d^3y}{dx^3}}-\sqrt[12]{\frac{d^2y}{dx^2}}=0\)
This matches (I) 6. Moving one term across and raising both sides to the power 12:
\( \displaystyle \left(\frac{d^3y}{dx^3}\right)^{1/2} = \left(\frac{d^2y}{dx^2}\right)^{1/12} \)
\( \displaystyle \left(\frac{d^3y}{dx^3}\right)^{6} = \frac{d^2y}{dx^2} \)
The highest order derivative \(\frac{d^3y}{dx^3}\) appears with power 6.
(C) \(\displaystyle\left(\frac{d^2y}{dx^2}\right)^2+\frac{dy}{dx}+e^{\frac{dy}{dx}}=x^2\)
This matches (II) Not defined. The term \(e^{dy/dx}\) cannot be written as a polynomial in \(\frac{dy}{dx}\), so the degree is not defined.
(D) \(\displaystyle\sqrt[3]{\frac{dy}{dx}}-\frac{d^2y}{dx^2}=e^x\)
This matches (III) 3. Isolating the radical and cubing:
\( \displaystyle \sqrt[3]{\frac{dy}{dx}} = \frac{d^2y}{dx^2} + e^x \)
\( \displaystyle \frac{dy}{dx} = \left(\frac{d^2y}{dx^2} + e^x\right)^3 \)
The highest order derivative \(\frac{d^2y}{dx^2}\) appears with power 3.
So the matching is (A)-(IV), (B)-(I), (C)-(II), (D)-(III).
Tip: Remove the radicals before reading off the degree. In (B) the powers look like \(\frac{1}{2}\) and \(\frac{1}{12}\), but the degree is 6.