Question

The integrating factor of \( y + \frac{d}{dx}(xy) = x(\sin x + \log x) \) is

• A. \( x \)
• B. \( \log x^2 \)
• C. \( x^2 \)
• D. \( x^3 \)

Hint:

$e^{\log(f(x))} = f(x)$. Always simplify the exponent in I.F.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
Expand the derivative term first to identify the standard linear form.
Step 2: Key Formula or Approach:
For \( \frac{dy}{dx} + P(x)y = Q(x) \), the integrating factor is \( IF = e^{\int P(x) dx} \).
Step 3: Detailed Explanation:
Expand \( \frac{d}{dx}(xy) = x \frac{dy}{dx} + y \).
The equation becomes:
\[ y + x \frac{dy}{dx} + y = x(\sin x + \log x) \] \[ x \frac{dy}{dx} + 2y = x(\sin x + \log x) \] Divide by \( x \):
\[ \frac{dy}{dx} + \frac{2}{x} y = \sin x + \log x \] Here \( P(x) = \frac{2}{x} \).
\[ IF = e^{\int \frac{2}{x} dx} = e^{2 \log x} = e^{\log x^2} = x^2 \] Step 4: Final Answer:
The integrating factor is \( x^2 \).