Question

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

• A. \( \frac{1}{8x} \)
• B. \( e \)
• C. \( e^{\log x} \)
• D. \( \log x \)

Hint:

For solving differential equations, the integrating factor often simplifies the equation by removing non-homogeneous terms. Check the form of the equation and use standard methods to find the integrating factor.

Answer 1

The Correct Option is D

The given equation is of the form: \[ \frac{dx}{dy} = \frac{x \log x}{2 \log x - y} \] To find the integrating factor, we need to identify a function that will multiply the entire equation to make it easier to solve. In this case, the integrating factor is determined by identifying the term that simplifies the equation when multiplied by \( x \), leading to a solvable equation. Through the process of solving such equations, we find the integrating factor to be \( \log x \).