Question

The integrating factor of the differential equation \( 2x \frac{dy}{dx} - y = 3 \) is

• A. \( \sqrt{x} \)
• B. \( \frac{1}{\sqrt{x}} \)
• C. \( e^x \)
• D. \( e^{-x} \)

Hint:

Always ensure the coefficient of \( \frac{dy}{dx} \) is 1 before identifying \( P(x) \).
Remember logarithmic properties: \( n \log a = \log a^n \) and \( e^{\log m} = m \).

Answer 1

The Correct Option is B

Concept:
• A linear differential equation of the first order has the standard form \( \frac{dy}{dx} + P(x)y = Q(x) \).
• The Integrating Factor (IF) is calculated using the formula: \( IF = e^{\int P(x) dx} \).

Step 1: Convert the given equation to the standard form
The given equation is \( 2x \frac{dy}{dx} - y = 3 \).
Divide the entire equation by \( 2x \) to make the coefficient of \( \frac{dy}{dx} \) equal to 1:
\[ \frac{dy}{dx} - \frac{y}{2x} = \frac{3}{2x} \]
\[ \frac{dy}{dx} + \left( -\frac{1}{2x} \right)y = \frac{3}{2x} \]

Step 2: Identify \( P(x) \)
Comparing with the standard form \( \frac{dy}{dx} + P(x)y = Q(x) \):
\[ P(x) = -\frac{1}{2x} \]

Step 3: Calculate the Integrating Factor (IF)
\[ IF = e^{\int P(x) dx} = e^{\int -\frac{1}{2x} dx} \]
Take the constant out of the integral:
\[ IF = e^{-\frac{1}{2} \int \frac{1}{x} dx} \]
Using \( \int \frac{1}{x} dx = \log x \):
\[ IF = e^{-\frac{1}{2} \log x} = e^{\log x^{-1/2}} \]

Step 4: Simplify the expression
Using the property \( e^{\log f(x)} = f(x) \):
\[ IF = x^{-1/2} = \frac{1}{x^{1/2}} = \frac{1}{\sqrt{x}} \]
This matches option (B).