Question

An integrating factor of the differential equation \( \sin x \frac{dy}{dx} + 2y \cos x = 1 \) is:

• A. \( \sin^2 x \)
• B. \( \frac{2}{\sin x} \)
• C. \( \log |\sin x| \)
• D. \( \frac{1}{\sin^2 x} \)
• E. \( 2 \sin x \)

Hint:

Standardizing the coefficient of $dy/dx$ to 1 is the most critical first step. Forgetting to divide by the leading term is the most common error in finding integrating factors.

Answer 1

The Correct Option is A

Concept: For a linear differential equation in standard form \( \frac{dy}{dx} + P(x)y = Q(x) \), the integrating factor (I.F.) is given by \( e^{\int P(x) dx} \).

Step 1: Convert to standard form.
Divide the entire equation by \( \sin x \): \[ \frac{dy}{dx} + \frac{2 \cos x}{\sin x} y = \frac{1}{\sin x} \] \[ \frac{dy}{dx} + (2 \cot x)y = \csc x \]

Step 2: Identify \( P(x) \).
Here, \( P(x) = 2 \cot x \).

Step 3: Calculate the Integrating Factor.
\[ \text{I.F.} = e^{\int 2 \cot x \, dx} \] We know that \( \int \cot x \, dx = \log|\sin x| \): \[ \text{I.F.} = e^{2 \log|\sin x|} = e^{\log(\sin^2 x)} \] Using the property \( e^{\log z} = z \): \[ \text{I.F.} = \sin^2 x \]