91Ó°ÊÓ

A first-order linear differential equation is a relationship between a function and its first derivative. It takes the form \(y' + p(x)y = q(x)\), where \(p(x)\) and \(q(x)\) are functions of \(x\).
In these equations, the highest order derivative is the first derivative, hence the term 'first-order'. The equation we have, \(y' = x - y\), fits this form once rearranged to \(y' + y = x\).
This type of differential equation is called linear because the function \(y\) and its derivative appear to the power of one, and purely in a linear form. They are important in mathematical modeling of systems and phenomena because they can describe a wide range of physical processes.

An integrating factor is a crucial method for solving linear first-order differential equations. It is typically denoted as \(\mu(x)\) and is used to simplify the equation.
To find the integrating factor, we compute \(\mu(x) = e^{\int p(x) \, dx}\), where \(p(x)\) is the coefficient of \(y\) in the standard form.
For the equation \(y' + y = x\), we identify \(p(x) = 1\). Thus, the integrating factor is \(e^x\).
By multiplying the entire differential equation by this factor, the left-hand side becomes an exact derivative, allowing straightforward integration. This simplifies the approach to finding a solution.

Integration by parts is a technique to integrate products of functions. It is based on the formula \[\int u \, dv = uv - \int v \, du\]. This method is particularly handy when dealing with an integral like \(\int e^x x \, dx\).
In the process for solving such integrals, we choose parts: \(u = x\) and \(dv = e^x \, dx\). Then, differentiate and integrate respectively to find \(du = dx\) and \(v = e^x\).
Substituting these into the formula, we split the original integral into simpler parts and compute: \(e^x x - \int e^x \, dx\). Solving this leads to \(e^x x - e^x + C\), which is key in solving the differential equation.

The general solution of a differential equation is an expression that encompasses all possible solutions. For first-order linear equations, the general solution includes a constant \(C\), representing any particular solution dependent on initial conditions.
In our problem, after solving the differential equation \(y' = x - y\) using integrating factors and integration by parts, we arrive at \(y = x - 1 + Ce^{-x}\).
Here, \(x - 1\) is the particular solution to the non-homogeneous part, and \(Ce^{-x}\) is the solution to the homogeneous part (when \(x = 0\)).
The constant \(C\) allows for adjustment to specific initial conditions, ensuring the solution fits given criteria in a real-world context.