Differential Equations I

Question 1

Define a Lipschitz function

Answer 1

A function f : R2 → R, satisfies a Lipschitz condition (with Lipschitz constant L) if there exists L > 0 such that |f(x, y1) − f(x, y2)| ≤ L|y1 − y2| for all x ∈ R and all y1, y2 ∈ R.

Question 2

State Picard’s Theorem

Answer 2

Let f : R → R be a function defined on the rectangle R := {(x, y) : |x − a| ≤ h, |y − b| ≤ k} which satisfies:
P(i): f is continuous in R with |f(x, y)| ≤ M for all (x, y) ∈ R, where Mh ≤ k
P(ii): f satisfies a Lipschitz condition in R.
Then the IVP y’(x) = f(x, y(x)) with y(a) = b. has a unique solution y on the interval [a-h, a+h].

Question 3

State Gronwall’s Inequality

Answer 3

Suppose A,B ≥ 0 are constants and v is a non-negative continuous function satisfying v(x) ≤ B + A|integral from x to a of v(s)| for all x in interval [a − h1, a + h2], h1,h2 ≥ 0. Then v(x) ≤ Be^(A|x−a|) for all x ∈ [a − h1, a + h2].

Question 4

State the comparison principle

Answer 4

Let g: R2 → R be continuously differentiable and let u(x) and v(x) be differentiable functions which satisfy u’(x) ≤ g(x, u(x)) and v’(x) ≥ g(x, v(x)) on some interval I. If u(x0) ≤ v(x0) for some x0 ∈ I then we obtain that u(x) ≤ v(x) for all x ∈ I with x ≥ x0 while if u(x0) ≥ v(x0) then we obtain that u(x) ≥ v(x) for all x ∈ I with x ≤ x0.