APPM 2360 Midterm: appm2360fall2013exam1
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If the solution oscillates, this means that isoclines would need to depened on time and thus the di erential equation would not be autonomous. Problem 2: (30 points) consider the di erential equation. Solution: (a) the only equilibrium is y = 0. If y 6= 0, we can use separation of variables to rewrite the ode as (2) (t + 1) 2/3dt = y 1/3dy. 2 2 + c c = 0. (3) (4) Therefore the solution to the ivp is given by. For (i), f is not continuous at (t, y) = ( 1, 1), so the solution may not exist and may be non-unique. For (ii), f is continuous but f / y is not continuous at (t, y) = (0, 0), so the solution exists but may be non-unique. For (iii), both f and f / y are continuous at (t, y) = (0, 1), so the solution exists and is unique.
