Final Answer:
The solution to the given exact differential equation is y⁴ - 2ty² = C, where C is the arbitrary constant.
Step-by-step explanation:
In this exact differential equation, the goal is to find a function y(t) such that the expression involving t, y, and dy/dt is the total differential of some function with respect to both t and y. The given equation is 2t - y - (2y⁻³ + t)dy/dt = 0. To find the solution, integrate the expression with respect to y, treating t as a constant. The integral yields y⁴ - 2ty² + g(t), where g(t) is a function of t alone. This expression is the potential function for the exact differential equation.
Now, to determine g(t), take the partial derivative of the potential function with respect to t and set it equal to the remaining term involving t in the original equation. The partial derivative is -2y² + g'(t) = 2t. Solve for g'(t), and integrate it with respect to t to find g(t). Substituting g(t) back into the potential function, the final solution is y⁴ - 2ty² = C, where C is the constant of integration.
This approach ensures that the solution satisfies the conditions for exactness and provides a clear understanding of the steps involved in finding the solution to an exact differential equation.