The Computable Differential Equation Lecture ... - Bruce E. Shapiro

10 CHAPTER 1. CLASSIFYING THE PROBLEM
D,
√
K|y 1 − y 2 | ≥ |f(y 1 , y 2 )| =
∣
√4 − y1 2 − 4 − y2
2 ∣ (1.37)
Let y 2 = 2 and y 1 = 2 − ɛ for some small number ɛ. Then
K|ɛ| ≥ ∣ √ ∣ ∣∣
4 − (2 − ɛ) 2 (1.38)
∣
≥ ∣√
4ɛ − ɛ 2
∣ (1.39)
K 2 ɛ 2 ≥ 4ɛ − ɛ 2 (1.40)
(K 2 + 1)ɛ 2 ≥ 4ɛ (1.41)
K 2 + 1 ≥ 4 ɛ
(1.42)
K 2 ≥ 4 ɛ − 1 (1.43)
But since we can choose ɛ to be arbitrarily small (including 0), the right hand side of
the equation can be arbitrarily large. But then K is not a finite number, especially
when ɛ = 0. So f(t, y) is not Lipshitz, either. Again, this does not guarantee
non-uniqueness; it just tells us that uniqueness is not guaranteed.
1.4 Boundary Value Problems
Definition 1.6 (Boundary Value Problem). Let Let D ∈ R n+1 be a set. Let
(t 0 , y 0 ) ∈ D and suppose that f(t, y) : D ↦→ R n . Then
y ′ = f(t, y) (1.44)
g(y(0), y(b)) = 0 (1.45)
for some function g on R 2n is called a boundary value problem (BVP). The
constraint 1.45 is called a boundary condition.
In a typical boundary problem is one where the solution is constrained at both
ends of the interval. A scalar first order boundary problem would thus be overdetermined;
hence any BVP is by necessity a first order vector system (see box).
Example 1.5. Solve the boundary value problem
y ′′ + ω 2 y = 0 (1.46)
y(0) = 1 (1.47)
y(2π) = 1 (1.48)
Solution. We observe that any linear combination of sin and cos functions satisfies
the differential equation:
y = a cos t + b sin t (1.49)
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007