Solve First Order Linear Differential Equation2

Solve the differential equation: $\frac{dy}{dx} - y = x$.

This problem involves solving a first order linear differential equation, a fundamental concept in differential equations. Solving such equations often requires the use of an integrating factor, a technique that simplifies the equation into a more manageable form. For these types of problems, identifying the structure of the differential equation is key. Recognize that the given equation can be expressed in the standard form of a linear differential equation, which allows you to apply the integrating factor method.

This technique involves multiplying through the entire equation by an integrating factor, which is usually an exponential function derived from the coefficient of y in the equation. By doing this, the left-hand side becomes the derivative of the product of the unknown function and the integrating factor, allowing you to integrate both sides easily.

Understanding this concept is crucial as it is a stepping stone to more complex differential equations encountered in higher mathematics courses. Moreover, solving linear differential equations is not just an academic exercise—these equations have applications in modeling real-world phenomena such as population dynamics, heat transfer, and financial mathematics. Thus, mastering this method not only equips you with the theoretical knowledge but also prepares you to tackle practical problems using mathematical modeling.

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