1. Solve the following initial value problem:
y'= 1-y², y(0) = 0.
2. Solve the initial value problem
y''- y - 2y = 0, y(0) = a, y'(0) = 2.
Then find a so that the solution approaches 0 as t? ?.
3. Solve the following initial value problem:
y'' - 2y' + 5y = 0, y(?/2) = 0, y'(?/2) = 2.
Sketch the graph of the solution and describe its behavior for increasing t.
4. Consider the initial value problem for y = y(t), t ? 0:
y' = ?1-y², y(0) = -1.
Show that there are infinitely many solutions to the above initial value problem.
