Question

The integrating factor of differential equation \( R \frac{dx}{dy} + Px = Q \) where \( P, Q, R \) are functions of \( y \) is

• A. \( e^{\int \frac{P}{Q} dy} \)
• B. \( e^{\int P dy} \)
• C. \( e^{\int \frac{P}{R} dy} \)
• D. \( e^{\int \frac{P}{R} dx} \)

Hint:

Always ensure the coefficient of the derivative term (\( \frac{dx}{dy} \) or \( \frac{dy}{dx} \)) is 1 before identifying \( P \).
The integral in the exponent must be with respect to the independent variable (\( y \) in this case).

Answer 1

The Correct Option is C

Concept:
• A first-order linear differential equation in \( x \) has the standard form: \[ \frac{dx}{dy} + P_1(y)x = Q_1(y) \]
• The Integrating Factor (I.F.) for such an equation is defined as: \[ \text{I.F.} = e^{\int P_1(y) dy} \]

Step 1: Convert the given equation to standard form
The given equation is: \[ R \frac{dx}{dy} + Px = Q \] To get it into standard form, we divide the entire equation by \( R \) (assuming \( R \neq 0 \)): \[ \frac{dx}{dy} + \left( \frac{P}{R} \right) x = \frac{Q}{R} \]

Step 2: Identify the coefficient of \( x \)
Comparing with the standard form \( \frac{dx}{dy} + P_1 x = Q_1 \), we find: \[ P_1 = \frac{P}{R} \] Note that \( P \) and \( R \) are given as functions of \( y \), so \( P_1 \) is also a function of \( y \).

Step 3: Calculate the Integrating Factor
Using the formula for I.F.: \[ \text{I.F.} = e^{\int P_1 dy} \] \[ \text{I.F.} = e^{\int \frac{P}{R} dy} \] This matches option (C).