Final answer:
To solve the given initial value problem, we separate variables and integrate the rearranged first-order differential equation, then apply the initial condition y(1) = 3 to find the particular solution.
Step-by-step explanation:
To solve the initial value problem (3y² - t/y⁵) dy/dt + t/2y⁴ = 0, where y(1)=3, we need to separate the variables and integrate both sides. Since this is a first-order differential equation, we will express it in the form of dy/dt = f(y, t) to make it possible for separation of variables.
First, separate the y and t parts of the equation:
- Multiply through by y⁵ to get 3y⁷ dy - t dy + ½t y dt = 0.
- Rearrange terms to group y's and t's: (3y⁷ - ½t) dy + t y dt = 0.
- Divide through by y(3y⁶ - ½t) to isolate dt on one side: dy / (3y⁶ - ½t) = -t dt / y.
- Integrate both sides with respect to their variables.
The solution will involve finding the antiderivatives of the separated equation, then solving for the constant of integration using the initial condition y(1) = 3. Finally, we will solve for y as a function of t.