Question

An integrating factor of the differential equation \[ x \, dy + (1 - y) \, dx = 0 \] is

• A. \( \frac{1}{x} \)
• B. \( x \)
• C. \( e^x \)
• D. \( \frac{1}{x^2} \)

Hint:

When you encounter a separable differential equation, an integrating factor can often be identified by looking at the form of the terms. In this case, multiplying by \( \frac{1}{x} \) simplifies the equation for integration.

Answer 1

The Correct Option is A

The given differential equation is: \[ x \, dy + (1 - y) \, dx = 0. \] Rearranging this equation to group similar terms on both sides, we get: \[ x \, dy = (y - 1) \, dx. \] Next, divide both sides of the equation by \( x(y - 1) \), which results in: \[ \frac{dy}{y - 1} = \frac{dx}{x}. \] This equation is separable, meaning we can integrate both sides independently. To do so, we can multiply through by an integrating factor. Notice that the term \( \frac{dx}{x} \) suggests that the integrating factor should be something that makes the equation easier to handle. By inspection, we identify that multiplying through by \( \frac{1}{x} \) would simplify the equation significantly. So, the integrating factor is \( \frac{1}{x} \), which makes both sides integrable directly. Thus, the integrating factor required for this equation is \( \frac{1}{x} \), making option (A) the correct answer.