what is the integration facto of dy / dx + 1 / x   y= x square

To solve the differential equation \( \frac{dy}{dx} + \frac{1}{x} y = x^2 \), we can identify that this is a first-order linear differential equation of the form:

\[
\frac{dy}{dx} + P(x)y = Q(x)
\]

Here:
- \( P(x) = \frac{1}{x} \)
- \( Q(x) = x^2 \)

### Step 1: Finding the integrating factor
The integrating factor \( \mu(x) \) is given by:

\[
\mu(x) = e^{\int P(x) dx} = e^{\int \frac{1}{x} dx}
\]

The integral of \( \frac{1}{x} \) is \( \ln |x| \), so:

\[
\mu(x) = e^{\ln |x|} = |x|
\]

Since \( x \) is positive (due to the form of the problem), we can write:

\[
\mu(x) = x
\]

### Step 2: Multiply the entire equation by the integrating factor
Now, multiply the original differential equation by \( x \):

\[
x \cdot \frac{dy}{dx} + \frac{x}{x} y = x \cdot x^2
\]

Simplifying:

\[
x \frac{dy}{dx} + y = x^3
\]

### Step 3: Solve the equation
The left-hand side can now be written as a derivative:

\[
\frac{d}{dx} (xy) = x^3
\]

Now, integrate both sides with respect to \( x \):

\[
xy = \int x^3 dx
\]

The integral of \( x^3 \) is:

\[
xy = \frac{x^4}{4} + C
\]

### Step 4: Solve for \( y \)
Now, divide by \( x \):

\[
y = \frac{x^3}{4} + \frac{C}{x}
\]

Thus, the solution to the differential equation is:

\[
y = \frac{x^3}{4} + \frac{C}{x}
\]

Would you like further clarification or more details on the process? Here are five related questions you might explore:

1. How do we handle the integration factor for more complex functions?
2. What happens if \( P(x) \) is a more complicated function, such as a trigonometric one?
3. How does the integrating factor method compare to other methods for solving first-order linear differential equations?
4. Can this method be extended to non-linear differential equations?
5. How would the solution change if \( Q(x) \) was \( e^x \) instead of \( x^2 \)?

**Tip:** Always verify the final solution by plugging it back into the original differential equation.

what is the integration facto of dy / dx + 1 / x y = x square

This page provides a step-by-step solution for solving the first-order differential equation dy/dx + (1/x)y = x². Learn how to apply the integrating factor method to find the general solution, covering key calculus concepts such as integration and linear differential equations. Ideal for undergraduate students or advanced high school calculus learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation