Creation Phase. Solve the following problems, clearly outlining and explaining your thought
process, in such a way that you and your peers can easily understand your solutions. You can
either write very neatly on paper, and scan your solutions with your phone (use the Dropbox app
rather than just a photo,) or (even better) type your solutions (using for exemple Overleaf, which
is described in Pip’s Tutorial 0,) and upload to your Kritik account. This step is due by 11:00pm
on Wednesday Oct 6th.
Consider the initial value problem (at time t0 = 1):
(
x˙ = f(x) = (x − a)
1/n, a ∈ R, n > 1 odd,
x(1) = x0
(A) Find the largest set A ⊂ R where f is continuous, and the largest set B ⊂ R where f is
continuously differentiable.
(B) Using the Existence and Uniqueness Theorem from class, find all possible initial values x0 at
t0 = 1 for which the above initial value problem has a unique solution in some time interval around
t0 = 1.
(C) For the remaining initial values x0 at t0 = 1, find at least 4 different solutions.
(D) Finally, consider the special case a = 1, n = 3 and x0 = 0, namely the following initial value
problem:
(
x˙ = (x − 1)1/3
,
x(1) = 0
Using (C), explain why there is a unique solution to this problem in some time interval around
t0 = 1. Find that solution explicitly and find the largest time interval I ∈ R in which this solution
makes sense. This solution can be extended to all of R: find an extension and explain why this
extension is not unique. (Hint: recall (A).)
Evaluation Phase. After the due date you will receive 4 submissions of solutions to the above
problems, which you are to evaluate using the criteria in the rubric: Results, Explanation, Interpretation and Presentation. The written form of the solution and a clear explanation of the approach
used is as important as the mathematical content. The idea is to see how your classmates solve
and present the results, and to give feedback to your peers. You will be required to make a written
criticism of each submission, indicating places where the student has done well and where improvement is needed. (You will be evaluated on how useful your remarks are!) The ideal is what you
would expect in a textbook: a complete and perfectly written solution which explains everything in
a concise and clear way. You will receive a solution which you may use to help you in the evaluation
phase. Remember that there may be more than one correct and clear method for proving any
statement in mathematics
Sample Solution