Question

The integrating factor of the differential equation $(1 + x^2) \frac{dy}{dx} + 2xy = \cos x$ is:

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

Hint:

If the numerator is the exact derivative of the denominator, the integral is simply the natural log of the denominator, yielding a direct simplification with the base $e$.

Answer 1

The Correct Option is B

Step 1: Concept
To find the integrating factor of a linear differential equation, we first write it in standard form: $\frac{dy}{dx} + P(x)y = Q(x)$. The integrating factor is $\text{I.F.} = e^{\int P(x) \, dx}$.

Step 2: Meaning
We divide the entire equation by $(1 + x^2)$ to isolate $\frac{dy}{dx}$ and identify the function $P(x)$.

Step 3: Analysis
Dividing by $(1 + x^2)$: \[ \frac{dy}{dx} + \left(\frac{2x}{1 + x^2}\right)y = \frac{\cos x}{1 + x^2} \] Thus, $P(x) = \frac{2x}{1 + x^2}$. Now calculate the integrating factor: \[ \text{I.F.} = e^{\int \frac{2x}{1 + x^2} \, dx} \] Using $u$-substitution for the exponent integration ($\int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)|$): \[ \text{I.F.} = e^{\ln(1 + x^2)} = 1 + x^2 \]

Step 4: Conclusion
The integrating factor is $1 + x^2$, which corresponds to option (B).

Final Answer: (B)