Question

Match List-I with List-II:

• A. (A)- (I), (B)- (III), (C)- (IV), (D)- (II)
• B. (A)- (I), (B)- (IV), (C)- (III), (D)- (II)
• C. (A)- (II), (B)- (I), (C)- (III), (D)- (IV)
• D. (A)- (III), (B)- (IV), (C)- (II), (D)- (I)

Hint:

When solving differential equations, an integrating factor is often needed to make the equation exact. The choice of integrating factor depends on the form of the equation. For instance, if the equation involves terms that suggest a power of \( x \) or \( y \), you may need to multiply by a factor such as \( x^n \) or \( y^m \). Recognizing patterns and knowing how the integrating factor affects the equation can help simplify the process.

Answer 1

The Correct Option is B

For equation (A) \(xdy - (y + 2x^{-2})dx = 0\), the integrating factor is dependent on \(\frac{1}{x}\). This corresponds to match (I).

For equation (B) \((2x^2 - 3y)dx = xdy\), the integrating factor is dependent on \(x^3\). This corresponds to match (IV).

For equation (C) \((2y + 3x^2)dx + xdy = 0\), the integrating factor is proportional to \(x^2\). This corresponds to match (III).

For equation (D) \(2xdy + (3x^3 + 2y)dx = 0\), the integrating factor is proportional to \(x\). This corresponds to match (II).

The matches are as follows: (A) – (I), (B) – (IV), (C) – (III), (D) – (II).