Integrating Factor Method for First Order Linear Differential Equations

Solve the first-order linear differential equation $y' + \frac{1}{x}y = 6x$ after rewriting it in linear form, using the integrating factor.

First-order linear differential equations are a fundamental topic in differential equations studies, serving as a building block for more complex types of differential equations. The given problem involves using the method of integrating factors to solve such an equation. Essentially, these problems require identifying the right factor that, when multiplied with the original differential equation, transforms it into a form that simplifies integration and eventual solving.

The integrating factor method revolves around the concept of identifying a multiplier— in this case, typically an exponential function— that helps reformulate the differential equation into an integrable format. Recognizing patterns and identifying the structure of the equation, such as isolating $y'$ and $y$ terms from non-$y$ terms, is critical. Knowledge of integration techniques and algebraic manipulation are fundamental skills needed to apply this method effectively.

After determining the appropriate integrating factor, solving the differential equation involves a series of straightforward steps, which should be followed diligently. Solving these kinds of problems enhances one’s ability to handle more complex equations and systems, building a deeper understanding of mathematical modeling processes.