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An integrating factor is a crucial tool for solving first-order linear differential equations. It is used to transform a non-exact differential equation into an exact one, enabling easier integration. The goal is to simplify the equation into a form where the derivative of a product appears, allowing for straightforward integration.
To find the integrating factor, consider a general first-order linear differential equation:

• \( y^{\prime} + P(x) y = Q(x) \)

Here, \( P(x) \) is a function of \( x \). The integrating factor \( \mu(x) \) is defined as:\[\mu(x) = e^{\int P(x) \, dx}\]By multiplying the entire differential equation by \( \mu(x) \), the equation becomes exact, making it easier to find a solution. In our problem, \( P(x) = -\frac{1}{2} \), so the integrating factor is:\[\mu(x) = e^{-\frac{1}{2}x}\]Using this integrating factor, the differential equation becomes more manageable and transforms into a derivative of a product.

Finding the general solution to a differential equation involves determining a function or a set of functions that satisfy the equation. After applying the integrating factor, the equation is now in a form:\[(e^{-\frac{1}{2}x} y)^{\prime} = \frac{1}{2}\]The left side of the equation appears as a derivative of a product, which is a key indicator that integration can be performed straightforwardly. Integrating both sides gives:\[e^{-\frac{1}{2}x} y = \frac{x}{2} + C\]where \( C \) is an integration constant. This process transforms the differential equation into the general solution form. To express \( y \) explicitly, simply solve for \( y \) by multiplying both sides by \( e^{\frac{1}{2}x} \):

• \( y = e^{\frac{1}{2}x} \left( \frac{x}{2} + C \right) \)

This is the general solution and represents all possible solutions to the differential equation, encompassing different scenarios depending on \( C \).

The integration constant, represented as \( C \) in the general solution, is a vital addition when integrating both sides of a differential equation. When integrating, the constant accounts for all possible initial conditions and scenarios that may be applied to the general solution.
Every differential equation represents a family of solutions, where each specific solution within that family corresponds to a particular initial condition or integration constant. For example, in the general solution:\[ y = e^{\frac{1}{2}x} \left( \frac{x}{2} + C \right)\]the integration constant \( C \) allows this equation to adapt to different initial values by providing a unique result for each value of \( C \).
Determining \( C \) often depends on additional information, such as initial conditions provided within a specific problem setting.