Solving a First Order Linear Differential Equation with Integrating Factors

Solve the differential equation: $\frac{dy}{dx} + \frac{2x}{1 + x^2} y = 4(1 + x^2)^2$ using an integrating factor.

In this problem, you are asked to solve a first order linear differential equation using the integrating factor method, a powerful technique for tackling linear differential equations of the form dy/dx + P(x)y = Q(x). Understanding the conceptual foundation of integrating factors is crucial, as it allows you to transform the differential equation into a form that is easier to solve. The core idea is to find a function, called the integrating factor, that when multiplied by the original differential equation, enables the left-hand side to be expressed as the derivative of a product. This simplifies the problem to finding antiderivatives, a far more straightforward task.

The given equation, with terms involving both x and trigonometric expressions, highlights the requirement for recognizing the pattern that facilitates the use of the integrating factor. The term involving y on the left-hand side hints at the standard form of a linear differential equation, while the right-hand side suggests how the problem might evolve after integrating. The key challenge lies in identifying the integrating factor, usually an exponential function derived from the coefficient of y. Once this factor is determined, multiplying through simplifies the integration process, reducing it to an algebraic manipulation.

Working through this type of problem reinforces an understanding of both integration techniques and the broader strategy for approaching linear differential equations. The ability to discern when and how to apply an integrating factor is a valuable problem-solving skill, offering insights into the symmetry and structure inherent in differential equations. This approach not only solves the equation, but also deepens your mathematical insight into the properties that govern such equations and their solutions.