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The integrating factor method is a powerful tool for solving first-order linear differential equations. This method transforms a non-exact equation into an exact one, allowing us to solve it more easily. A first-order linear differential equation often takes the form of \( y' + P(x)y = Q(x) \), where \( P(x) \) and \( Q(x) \) are functions of \( x \).
This equation becomes more manageable by employing the integrating factor, denoted \( \mu(x) \).

To determine the integrating factor:

• Identify the function \( P(x) \) in the equation.
• Find the integral of \( P(x) \), giving us \( \int P(x) \, dx \).
• The integrating factor is \( e^{\int P(x) \, dx} \).

In the example equation \((x+1) y^{\prime}+y=2 x\), dividing by \(x + 1\) puts it in the standard form, revealing \( P(x) = \frac{1}{x+1} \). The integral \( \int \frac{1}{x+1} \, dx = \ln|x+1| \), hence the integrating factor becomes \( \mu(x) = |x+1| \).
Using this integrating factor, the equation is multiplied by \( \mu(x) \), facilitating an easier integration process by turning the left-hand side into a derivative.

A first-order linear differential equation should first be written in its standard form to apply methods like the integrating factor easily. The standard form is expressed as:
\[ y' + P(x)y = Q(x) \]
Here, \( y' \) is the derivative of \( y \) with respect to \( x \), and \( P(x) \) and \( Q(x) \) are coefficients that depend on \( x \) only.

When faced with an equation not in this form:

• Manipulate it algebraically to isolate \( y' \) and express the equation with a single \( y \) accompanied by its coefficient \( P(x) \).
• Identify and adjust the equation so that \( Q(x) \) is on the right-hand side.

In the equation \((x+1) y^{\prime}+y=2 x\), division by \( x+1 \) makes it fit the standard form: \( y' + \frac{1}{x+1}y = \frac{2x}{x+1} \).
Achieving this form is crucial to facilitate the correct application of solving techniques, especially for using the integrating factor method effectively.

Once a first-order linear differential equation is expressed in standard form and solved via the integrating factor, finding the general solution is the final target. This solution represents a family of curves that embody all potential specific solutions under varying constants.

Steps to obtain the general solution:

• Multiply the standard equation by the integrating factor, making the left side integrable as a derivative of a product.
• Integrate both sides of the equation accordingly.
• Separate \( y \) from the rest of the expression for the final solution.

In the solved example, the integration of both sides gave \((x+1)y = x^2 + C\), where \( C \) is the integration constant. Solving for \( y \) by dividing both sides by \( x+1 \), we find:
\[ y = \frac{x^2 + C}{x+1} \]
Each value of \( C \) results in a unique solution, depicting the infinite possibilities woven into the general solution. Understanding this wraps up the solving process and provides complete insight into the differential equation's behavior.