Our professor gave us the following theorem:

Theorem: Let $\alpha = min\left(a, \dfrac{b}{M}\right)$ where $M = max |f|$ is the maximum of $|f|$ on $I \times J$ where $I= [x_0 - a, x_0 +a], J = [y_0 - b, y_0 +b]$. Then there exists a unique solution $\phi : I \rightarrow J$ f the Cauchy problem.

Now I encounter several problems when trying to understand this theorem:

  • $\alpha$ is defined at the beginning but isn't used in the theorem

  • $a$ is not defined

  • $b$ is not defined

Now, using this theorem I am supposed to find the interval on which there exists a solution to the following Cauchy Problem:$y' = 2y^2-x; y(1) =1$

So the only informmations I know are $(x_0, y_0) = (1,1)$

Therefore I can't use the theorem.

  • $\begingroup$ It seems "Then there exists a unique solution $ϕ:I→J$" must continue saying "...in the interval $[x-\alpha,x+\alpha]$ $\endgroup$ – Rafa Budría Mar 2 '17 at 16:28

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