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
ist’s “bacchantic howling only proves imcompetence”, who “merely vents [their feelings] in helplessly inarticulate sounds” . This seems to perfectly epitomise the character of Herr Friedemann, who drowns himself in water that is only deep enough to cover his face, leaving the rest of his body on the ground. Mann’s narration of the final scene encompasses the Apollonian and Dionysian psychology of the protagonist as he commits suicide. The free indirect style of “was ging eigentlich in ihm vor, bei dem, was nun geschah?” is delivered with a critical, cold impassivity that destroys Herr Friedemann’s possibly ‘Wagnerian’ dream of a love towards Frau von Rinnlingen, and recapitulates in his almost grotesque, mental annihilation that appears in stark contrast to the ‘Wilhemine mundanity’1 in which the novel is set. Mann provides a compelling narrative from a distinctly extra-diegetic standpoint. This removes all form of sympathy from his narration and enables the novel to become an example of social impropriety and moral redundancy; how can Herr Friedemann expect that Frau von Rinnlingen will suddenly become adulterous in this social setting, and hence how is it possible that he cannot be aware of the fact that this romantic passion that he feels is incongruous. This Dionysiac attitude is what creates his internal chaos; what I mean by this is that Herr Friedemann search for pleasure is deeply irrational and ultimately causes his pain. Nietzsche described such fusion of Dionysiac and Apollonian psychologies as ‘Kunsttriebe’, or ‘artistic impulses’, which form the basis of tragedy. We are able to apply this to Herr Friedemann, as his mental struggle arises as a result of his apparent love for ‘Kunst’, or rather his Apollonian search for aesthetic beauty, coupled with his natural ‘Trieb’, a certain Dionysiac helplessness that is presented through his desperate self-destruction at the end of novel. Mann may look to place Herr Friedemann’s suicide on nature, rather than an external factor, or the fault of Frau von Rinnlingen (she is certainly not to blame). The naturalist movement, that which Mann was a keen follower, grounded literature in scientific theory, whereby characterisation, hereditary disease, and psychology, are all linked by means of ‘biological determinism’. That is to say that Herr Friedemann’s fate is ultimately inevitable as a result of his degenerate life that he succumbs to, by being both physically and morally weaker. It is no coincidence that the hunchback he is fatefully given by the maid when he dropped in the first chapter, stunts his growth both in terms of stature and emotion. His misconstrued emotions therefore, tie in with his chaotic ‘Trieb’, that allows this naturalist, determinist work to ascertain an almost proleptic structure; Herr Friedemann anticipates and resolves potential problems, before they have even arisen; the finale tragically descends into an obdurate nihilistic dissection of his social misfortunes, that Mann presents through a naturalistic, fateful tone; an outsider’s voice that permeates through the free indirect style, commenting on what becomes, through Frau von Rinnlingen’s “kurzen, stolzen, verächtlichen Lachen”, a melodrama, that >
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