https://file.aiquickdraw.com/mathbot/file/1725973797995ylu492qv.jpg

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.

Which one of the following is the integrating factor of dy/dx + (1/x)y = x^2?

Explore how to determine the integrating factor of the first-order linear differential equation dy/dx + (1/x)y = x^2 using key concepts and formulas. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation