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 (A) \(xdy - (y + 2x^{-2})dx = 0\): The integrating factor here depends on \(\frac{1}{x}\). Match: (A) → (I).

For (B) \((2x^2 - 3y)dx = xdy\): The integrating factor here depends on \(x^3\). Match: (B) → (IV).

For (C) \((2y + 3x^2)dx + xdy = 0\): The integrating factor here is proportional to \(x^2\). Match: (C) → (III).

For (D) \(2xdy + (3x^3 + 2y)dx = 0\): The integrating factor is proportional to \(x\). Match: (D) → (II).

\((A) – (I), (B) – (IV), (C) – (III), (D) – (II)\).

Answer 2

For (A) \( xdy - (y + 2x^{-2})dx = 0 \):

The given equation is in the form: \[ xdy - \left(y + 2x^{-2}\right)dx = 0 \] To make this exact, we need an integrating factor that depends on \( \frac{1}{x} \). The correct match is **(A) → (I)**.

For (B) \( (2x^2 - 3y)dx = xdy \):

The equation is: \[ (2x^2 - 3y)dx = xdy \] The integrating factor here depends on \( x^3 \), which can be verified by simplifying the equation and checking for exactness. The correct match is **(B) → (IV)**.

For (C) \( (2y + 3x^2)dx + xdy = 0 \):

The equation is: \[ (2y + 3x^2)dx + xdy = 0 \] The integrating factor here is proportional to \( x^2 \). This is because multiplying through by an appropriate factor makes the equation exact. The correct match is **(C) → (III)**.

For (D) \( 2xdy + (3x^3 + 2y)dx = 0 \):

The equation is: \[ 2xdy + (3x^3 + 2y)dx = 0 \] The integrating factor here is proportional to \( x \), which makes the equation exact after multiplication by this factor. The correct match is **(D) → (II)**.

Conclusion:

The correct matches are:

• (A) → (I)
• (B) → (IV)
• (C) → (III)
• (D) → (II)