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Math 645: Homework 2
1. Show the following: If the Cauchy problem x0=f(t, x), x(t0) = x0, where f(t, x) is
a continuous function, has a unique solution, then the Euler polygons converge to
this solution.
2. Consider the Cauchy problem x0=f(t, x), x(0) = 0, where
f(t, x) =
4sign(x)q|x|if |x| ≥ t2
4sign(x)q|x|+ 4(t−|x|
t) cos(πlog t
log 2 ) if |x|< t2(1)
The function fis continuous on R2. Consider the Euler polygons xh(t) with h=
2−i,i= 1,2,3,· · ·. Show that xh(t) does not converge as h→0, compute its
accumulation points, and show that they are solution of the Cauchy problem. Hint:
the solutions are ±4t2.
3. Consider the the Cauchy problem x0=f(t, x), x(0) = 0 where fis given by
f(t, x) =
0 if t≤0, x ∈R
2tif t > 0, x ≤0
2t−4x
tif t > 0,0≤x<t2
−2tif t > 0, t2≤x
(2)
(a) Show that fis continuous. What does that imply for the Cauchy problem?
(b) Show that fdoes not satisfy a Lipschitz condition in any neighborhood of the
origin.
(c) Apply Picard-Lindel¨of iteration with x0(t)≡0. Are the accumulation points
solutions?
(d) Show that the Cauchy problem has a unique solution. What is the solution?
Note that this problem shows that existence and uniqueness of the solution does
not imply that the Picard-Lindel¨of iteration converges to the unique solution.
4. Consider the Cauchy problem x0=λx,x(0) = 1, with λ > 0 on the interval [0,1].
Compute the Euler polygons xh(t). Show that
λ
1 + λhxh(t)≤d
dtxh(t)≤λxh(t).(3)
and deduce from this the classical inequality
1 + λ
n!n
≤eλ≤ 1 + λ
n!n+λ
(4)
5. Let a,b,c, and dbe positive constants. Consider the Predator-Prey equation
x0=x(a−by), y0=y(cx −d) with positive initial conditions x(t0) and y(t0). Using
the change of variables p= log(x) and q= log(y) express the equations as an
Hamiltonian systems and deduce that the solution exist for all t.
