Question

Which of the following is an integrating factor for the differential equation $x \cos x \frac{dy}{dx} + (x \sin x + \cos x) y = 1$?

• A. $x \cos x$
• B. $x \sin x$
• C. $x \sec x$
• D. $x \csc x$

Hint:

For linear differential equations, the coefficient of $dy/dx$ must be 1 before calculating $P(x)$ and the Integrating Factor.

Answer 1

The Correct Option is C

Step 1: Concept
Convert to standard linear form $\frac{dy}{dx} + P(x)y = Q(x)$, then $IF = e^{\int P(x) dx}$.

Step 2: Meaning
Divide by $x \cos x$: $\frac{dy}{dx} + \frac{x \sin x + \cos x}{x \cos x} y = \frac{1}{x \cos x}$. So $P(x) = \frac{\sin x}{\cos x} + \frac{1}{x} = \tan x + \frac{1}{x}$.

Step 3: Analysis
$\int P(x) dx = \int (\tan x + \frac{1}{x}) dx = \log(\sec x) + \log x = \log(x \sec x)$.

Step 4: Conclusion
$IF = e^{\log(x \sec x)} = x \sec x$. Final Answer: (C)