MATH 557 Homework Set #4
Fall 2015
16. Suppose that h ∈ C(R)and assume f ∈ C(R2 ) satisfies |f (t, x)| ≤ h(t)|x| for all
(t, x) ∈ R2 . Show that for any (τ, ξ) the initial value problem
x0 = f (t, x)
x(τ ) = y
has a solution which exists for all t ∈ R.
17. Show that any solution of
x00 + x + x3 = 0
exists for all t ∈ R. (Since it is a second order equation, you will want to show that
neither of x(t), x0 (t) can become infinite at any finite time. Suggestion: multiply the
equation by x0 and use the resulting identity.)
18. The second order problem
x00 + λx − x2 = 0
x(τ ) = α x0 (τ ) = β
defines a solution function x = φ(t, τ, α, β, λ). Find an initial value problem satisfied by
the partial derivative of φ with respect to each of the parameters τ, α, β and λ. You may
calculate formally here, i.e. assume these derivatives exist and are sufficiently smooth.
You can either rewrite the equation as a first order system, or work directly with the
second order form.
19. For n = 1, 2, . . . let xn (t) denote the unique solution of
x0 =
n+
ntx
+ sin (t3 )
x2
x(0) = 1
Conjecture what x(t) = limn→∞ xn (t) should be, for 0 ≤ t ≤ 1 and prove your answer is
correct. (Suggestion: don’t try to solve for xn .)