Solving a Differential Equation with Trigonometric Expression

Solve for $y$ given $\frac{dy}{dx} = \frac{6x^2}{2y + \cos(y)}$.

This problem requires solving a first-order differential equation where the variables can be separated, making it a classic example of a separable differential equation. The given equation involves a rational expression with both polynomial and trigonometric components, and the goal is to isolate and integrate these components individually with respect to their respective variables.

In these types of problems, identifying the separable nature of the equation is crucial. The general approach involves rearranging the equation so that all terms involving y are on one side and all terms involving x are on the other. This separation allows you to integrate each side independently. The integration may involve slightly complex antiderivatives due to the presence of trigonometric functions, which are generally encountered in calculus courses.

Once the integration is performed, remember to include the integration constant, which will be determined if an initial condition is provided or left as is if not. In the context of applied problems, these constants often need to be solved for explicitly using given initial conditions, but when isolated, they serve to demonstrate the family of possible solutions to the differential equation.

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