Toronto Math Forum
MAT2442019F => MAT244Lectures & Home Assignments => Topic started by: Victor Ivrii on September 24, 2019, 10:56:17 AM

Answering question on discussions (https://q.utoronto.ca/courses/112004/discussion_topics/390029)
Definitely the role of the Existence and Uniqueness Theorems are much more important than Limits in Manual Computations (and honestly, I have no idea what "Manual Computation" means). However this role is more theoretical both in ODEs and PDEs. Still, if we are talking about numerical solutions (taught in different classes, we skip Chapter 8, and briefly look at section 2.7) we need to be pretty sure that the object we are trying to find exist and we find all of them.
For centuries from I. Newton (who introduced ODEs) mathematicians did not care much about existence, because they were looking for solutions of real life problems and believed in existence and also because the rigorous apparatus of Real Analysis which allows to prove such theorems came into existence only in 19th century. You may want to look at very sketchy Lecture_Note_to_Section_2.8_ExistenceUniqueness_Theorem (https://q.utoronto.ca/courses/112004/files/4002826?module_item_id=784500).
Uniqueness is a different matter: mathematicians observed that the solution to the Cauchy problem is not necessarily unique (remember, that the general solution to the 1st order ODE is $x=\varphi(t;C)$ or $\Phi(x, t; C)=0$ in the explicit and implicit form correspondingly and we need to specify one solution one needs to impose an extra condition; f.e. $x(t_0)=x_0$. They discovered that there could be a singular solution which is not a regular solution which means that it cannot be obtained from the general solution by freezing $C$ but which in each point coincides with some (depending on the point) regular solution. In more details see Lecture_Note_to_Chapter_2_Singular_Solutions (https://q.utoronto.ca/courses/112004/files/4151407?module_item_id=785534&fd_cookie_set=1).
Both of these lecture notes are optional