The question asks to find the integrating factor of the given first-order linear differential equation:

$\frac{dy}{dx} + \frac{1}{x}y = x^2$

General Form of a First-Order Linear Differential Equation:

A first-order linear differential equation has the general form: $\frac{dy}{dx} + P(x)y = Q(x)$ where $P(x)$ and $Q(x)$ are functions of $x$.

Integrating Factor:

The integrating factor ($\mu(x)$) is given by: $\mu(x) = e^{\int P(x) \, dx}$ where $P(x)$ is the coefficient of $y$ in the equation.

In this case, $P(x) = \frac{1}{x}$.

Calculation of Integrating Factor:

$\mu(x) = e^{\int \frac{1}{x} \, dx} = e^{\ln |x|} = x$

Thus, the integrating factor is $x$.

Answer:

The correct option is:

B) $x$.

Would you like more details or have any questions?

Related Questions:

1. How do you solve a differential equation after finding the integrating factor?
2. Can you explain how the integrating factor method works in general?
3. What happens if $P(x)$ is not $\frac{1}{x}$? How does that change the integrating factor?
4. What if the differential equation were non-linear? How would that affect solving it?
5. How can you verify that the integrating factor you found is correct?

Tip:

When solving a linear differential equation, always write it in the standard form first to easily identify $P(x)$ and apply the integrating factor formula correctly.