(Solution) - Discuss direction fields as follows a Graph a direction field -(2025 Original AI-Free Solution)

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Discuss direction fields as follows.
(a) Graph a direction field for the ODE y? = 1 ? y and in it the solution satisfying y(0) = 5 showing exponential approach. Can you see the limit of any solution directly from the ODE? For what initial condition will the solution be increasing, constant decreasing??
(b) What do the solution curves of y' = ? x3/y3 look like, as concluded from a direction field. How do they seem to differ from circles? What are the isoclines? What happens to those curves when you drop the minus on the right? Do they look similar to familiar curves? First, guess
(c) Compare, as best as you can, the old and the computer methods, their advantages and disadvantages,. Write a short report.