Question

Match List I with List II

List I	List II
\(A.\ [1 + (\frac{dy}{dx})^2] = \frac{d^2y}{dx^2}\)	I. order 2, degree 3
\(B. \ (\frac{d^3y}{dx^2})^2 - 3\frac{d^2y}{dx^2} + 2(\frac{dy}{dx})^4 = y^4\)	II. order 2, degree 1
\(C. \ (\frac{dy}{dx})^2 + (\frac{d^2y}{dx^2})^3 = 0\)	III. order 1, degree 2
\(D.\ (\frac{dy}{dx})^2 + 6y^3 = 0\)	IV. order 3, degree 2

Choose the correct answer from the options given below:

• A. А-І, В-II, C-IV, D-II
• B. А-III, В-I, C-II, D-IV
• C. A-IV, B-II, C-III, D-I
• D. A-II, B-IV, C-I, D-III

Answer 1

The Correct Option is D

To solve this problem, we need to match each differential equation in List I with its corresponding order and degree from List II. Let's analyze each equation:

A. \([1 + (\frac{dy}{dx})^2] = \frac{d^2y}{dx^2}\)
This equation represents a second-order differential equation because the highest derivative present is \(\frac{d^2y}{dx^2}\). The degree is 1 because the highest power of the highest order derivative is 1. Hence, this corresponds to List II: order 2, degree 1.

B. \((\frac{d^3y}{dx^2})^2 - 3\frac{d^2y}{dx^2} + 2(\frac{dy}{dx})^4 = y^4\)
The highest order derivative in this equation is \(\frac{d^3y}{dx^2}\), making it a third-order differential equation. Its degree is 2 because the highest order derivative \(\frac{d^3y}{dx^2}\) is raised to the power 2. Thus, it matches List IV: order 3, degree 2.

C. \((\frac{dy}{dx})^2 + (\frac{d^2y}{dx^2})^3 = 0\)
In this equation, the highest derivative is \(\frac{d^2y}{dx^2}\), indicating it is second-order. The degree here is 3 because \(\frac{d^2y}{dx^2}\) is raised to the power 3. This corresponds to List I: order 2, degree 3.

D. \((\frac{dy}{dx})^2 + 6y^3 = 0\)
For this equation, the highest derivative is \(\frac{dy}{dx}\), indicating first-order. The degree is 2, which is by the square of \(\frac{dy}{dx}\). Therefore, it matches List III: order 1, degree 2.

Based on this analysis, the correct matching is:
A-II, B-IV, C-I, D-III