Solving NonExact Differential Equations with Integrating Factors

Solve a non-exact differential equation using an integrating factor given the equation: $3x^2 y + x y^2 + x^3 + x^2 y$.

Non-exact differential equations are a type of first-order differential equation which cannot be solved through direct integration. Instead, these equations require us to use a technique to transform the equation into an exact form, which can then be integrated. The method of using integrating factors is a powerful technique for handling such situations. An integrating factor is a function, often derived from the non-exact equation, that when multiplied with the original equation renders it exact. The challenge lies in identifying the correct integrating factor and applying it properly.

In this problem, you will need to explore the relationship between the coefficients of the different terms in the equation to find an integrating factor. It often involves recognizing some patterns or using derivative relationships that make the equation's terms cohesive and integrate smoothly. Once the differential equation becomes exact, you can integrate it directly, which would yield the family of solutions to the original differential equation. Practicing this technique requires a good understanding of differentiable functions and experience with both exact and non-exact equations.

Exploring integrating factors is not only useful for theoretical purposes but also has practical applications in physics and engineering, where differential equations model many natural phenomena. The skill to convert a non-exact equation into an exact one extends your toolkit for solving complex real-world problems, largely expanding problem-solving strategies in mathematical modeling and equations.