Final answer:
To solve the exact differential equation, we verify exactness using the M and N functions, integrate with respect to the appropriate variables, and find the potential function whose level curves give us the general solution.
Step-by-step explanation:
To find the general solution for the given exact differential equation (DE), 2t - y - (2y^{-3} + t)(dy/dt) = 0, let's first verify its exactness and then solve it step by step.
First, we identify the function M(t, y) = 2t - y and the function N(t, y) = -(2y^{-3} + t) so that our DE looks like M(t, y) + N(t, y)*(dy/dt) = 0. We then confirm that ∂M/∂y = ∂N/∂t to ensure it's exact, which in this case should be true.
Next, we integrate M with respect to t and N with respect to y to find the potential function Ψ(t, y) such that ∂Ψ/∂t = M and ∂Ψ/∂y = N. Then the general solution will be given by the equation Ψ(t, y) = C, where C is a constant.
Following this procedure methodically will yield the general solution to the exact DE.
The process of finding and using the N and M functions, confirming the exactness, integrating, and finding the constant C are what help us arrive at the general solution for the DE.