Assignment Instructions/ Description
Image transcription text4. a) (8 pts) State the local Picard theorem for the initial value problem
dx
dt
= f(t, x), x(to) = TO-
Be sure to state the assumptions of the theorem clearly. Draw a diagram to illustrate the set-up of the theorem.
Explain how the size(s) of the region(s) in your diagram is/are chosen.... Show moreImage transcription text(Question 4 continued)
b) (7 pts) One of the assumptions of the theorem you stated in part a) implies a strong continuity property for the
function f(t, x); this property is used frequently throughout the proof. State what that property is, and explain
informally how it follows from the assumptions of the theorem.
b.2) (2 pts extra credit) By using the mean value theorem, prove rigorously that the strong continuity property of
f(t, x) follows from the assumptions of the theorem.... Show moreImage transcription text(Question 4 continued)
c) (5 pts) As we discussed in Lesson 10, the initial value problem
dx
= x', x(0) =1
has a unique solution, but that solution does not exist at t = 1. Explain carefully in what way this fact is consistent
with the local Picard theorem. Draw the appropriate diagram to illustrate your explanation. In your explanation,
state whether or not the assumptions of the theorem apply in this example, and prove your claim. In addition,
state whether the function in this example has the strong continuity property you defined in part b) and prove
your statement.... Show moreImage transcription text(Question 4 continued)
d) (5 pts) As we discussed in Lesson 10, the initial value problem
dx
dt
= 312/3, x(0) = 0
has infinitely many different solutions. Explain carefully in what way this fact is consistent with the local Picard
theorem. Draw the appropriate diagram to illustrate your explanation. In your explanation, state whether or
not the assumptions of the theorem apply in this example, and prove your claim. In addition, state whether the
function in this example has the strong continuity property you defined in part b) and prove your statement.... Show moreI need answer with brief explanation asap. Thanks!