General Solution of a FirstOrder Linear Differential Equation

Find the general solution of the first-order linear differential equation $\displaystyle \frac{dy}{dx} + \frac{2}{x} y = 3x - 5$.

First-order linear differential equations are foundational in the study of differential equations and appear frequently in various scientific fields. Solving these equations involves understanding both the structure of the equation and the method of integrating factors. The equation given has the form $\frac{dy}{dx} + P(x)y = Q(x)$, and finding its solution generally requires determining an integrating factor, which is a function that simplifies the equation into an exact differential equation. This allows us to express the solution in terms of an integral. In this particular problem, the integrating factor is dependent on the coefficient of y, specifically $\frac{2}{x}$, leading to certain simplifications.

Analyzing this problem helps in grasping the concept of linearity in differential equations and understanding how transformations can simplify complex expressions to more manageable forms. Understanding these techniques forms the basis for more advanced topics in differential equations, such as nonlinear differential equations and systems of differential equations. Such foundational skills are essential not just in further mathematical studies but also in physics, engineering, and other applied sciences where such differential equations model real-world phenomena like population dynamics, heat, and fluid flow.

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