First Order Linear Differential Equation with Integrating Factor

Solve the first-order linear differential equation rac{dy}{dx} - 2y = 6 using the integrating factor.

Solving first-order linear differential equations involves finding an integrating factor, a function that simplifies the differential equation into one that can be easily integrated. The strategy hinges on recognizing that the differential equation can be rewritten in a form that allows the derivative of the product of the function and the integrating factor to appear.

The integrating factor typically depends on the coefficient of the dependent variable, in this case 'y'. By selecting this factor appropriately, the left side of the equation becomes the derivative of a product, facilitating the integration process.

This method not only helps in solving a wide class of differential equations but also lays the groundwork for understanding higher-order linear differential equations tackled in advanced mathematics courses. It's also crucial for modeling real-world phenomena where rates of change are proportional to the value of a function as it evolves over time, such as in exponential growth and decay situations, electrical circuits, or fluid flow problems.