Question

The integrating factor of $x \cos x \frac{dy}{dx} + (x \sin x + \cos x) y = 1$ is}

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

Hint:

Always ensure the coefficient of $dy/dx$ is 1 before identifying $P(x)$ for the integrating factor formula.

Answer 1

The Correct Option is B

Step 1: Concept
The given equation is a first-order linear differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$. The Integrating Factor (I.F.) is calculated as $e^{\int P(x) dx}$.

Step 2: Meaning
First, we must normalize the equation by dividing the entire expression by the coefficient of $\frac{dy}{dx}$, which is $x \cos x$. This gives: $\frac{dy}{dx} + \frac{x \sin x + \cos x}{x \cos x} y = \frac{1}{x \cos x}$.

Step 3: Analysis
Here, $P(x) = \frac{x \sin x + \cos x}{x \cos x} = \frac{\sin x}{\cos x} + \frac{1}{x} = \tan x + \frac{1}{x}$. To find the I.F., we compute $e^{\int (\tan x + \frac{1}{x}) dx}$. The integral is $\ln(\sec x) + \ln x = \ln(x \sec x)$. Thus, I.F. $= e^{\ln(x \sec x)} = x \sec x$.

Step 4: Conclusion
Dividing by $x \cos x$ leads to $P(x) = \tan x + \frac{1}{x}$. The I.F. is $e^{\ln|x| + \ln|\sec x|} = x \sec x$. Final Answer: (B)