1. Let f : R ! R be a smooth function so that dxdnn f is bounded for n = 0; 1; 2 and consider the ODE
(
• = f(x)
x(0) = x0:
Let x1 be the approximation to x( t) obtained from the improved Euler method. Using Taylor’s Theorem, show that the local truncation error e1 = x( t) x1 satisfies
je1j C( t)3
for some constant C > 0.
2. Suppose that f : (a; b) ! R is Lipschitz. Show that f is continuous on (a; b). Solution
3. Let f : R ! R be Lipschitz and let x be a fixed point of the ODE
• = f(x):
Show that there cannot exist a solution with x(0) = x0 6= x that reaches the fixed point x in finite time.
Hint: Suppose for a contradiction that such a solution exists. What can you say about uniqueness?
4. (Exercise 2.5.3 in Strogatz) Consider the equation x = rx + x3, where r > 0 is fixed. Show that jx(t)j ! 1 in finite time, starting from any initial condition x0 6= 0.
5. Solve problem 2.5.2 in Strogatz.
6. Consider the equation
x = r +
1
x
x
:
4
1 + x
At what value of r do we have a saddle-node bifurcation?
