# Toronto Math Forum

## MAT244--2019F => MAT244--Lectures & Home Assignments => Topic started by: Victor Ivrii on September 24, 2019, 10:56:17 AM

Title: Existence and Uniqueness Theorem
Post by: Victor Ivrii on September 24, 2019, 10:56:17 AM
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).