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Match List-I with List-II
Choose the correct answer from the options given below: |
(A)-(I), (B)-(II), (C)-(III), (D)-(IV) (A)-(II), (B)-(III), (C)-(IV), (D)-(I) (A)-(IV), (B)-(III), (C)-(I), (D)-(II) (A)-(IV), (B)-(III), (C)-(II), (D)-(I) |
(A)-(IV), (B)-(III), (C)-(I), (D)-(II) |
The correct answer is Option (3) → (A)-(IV), (B)-(III), (C)-(I), (D)-(II) Given differential equations and to find order and degree: (A) \(\frac{d^2y}{dx^2} + 2 \left(\frac{dy}{dx}\right)^2 = e^{\frac{dy}{dx}} + 1\) - Order is highest derivative: \(\frac{d^2y}{dx^2}\) ⇒ Order = 2 - Degree is the power of highest order derivative after removing radicals and fractions of derivatives. Here, \(\frac{d^2y}{dx^2}\) is to the power 1 but RHS has exponential of \(\frac{dy}{dx}\), which is not a polynomial expression in derivatives ⇒ Degree is not defined. (B) \(\left(\frac{d^2y}{dx^2}\right)^2 + 4 \left(\frac{dy}{dx}\right)^3 = e^y - 1\) - Highest order derivative is \(\frac{d^2y}{dx^2}\) squared ⇒ Order = 2 - Degree is the power of highest order derivative after simplification ⇒ Degree = 2 (C) \(3 \frac{dy}{dx} + 4y + e^y = \frac{dy}{dx}\) - Highest order derivative is \(\frac{dy}{dx}\) ⇒ Order = 1 - Degree is the power of \(\frac{dy}{dx}\) after simplification. Rearranging: \(3 \frac{dy}{dx} - \frac{dy}{dx} = -4y - e^y \Rightarrow 2 \frac{dy}{dx} = \ldots\), degree of \(\frac{dy}{dx}\) is 1, no powers, but the term \(e^y\) is not involving derivative. But since \(\frac{dy}{dx}\) appears to power 1, degree = 1 (D) \(\frac{d^2y}{dx^2} + 3 \frac{dy}{dx} = (e^y + \frac{dy}{dx})^2\) - Highest order derivative is \(\frac{d^2y}{dx^2}\) ⇒ Order = 2 - Degree is power of \(\frac{d^2y}{dx^2}\) which is 1 (since it is not raised to any power) - The RHS is a square of expression involving first derivative, but highest order derivative is second derivative raised to 1 only ⇒ Degree = 1 Matching:
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