Task: I had to find out some estimates for M and L to make sure the proportional accucrazy is not above $10^{-4}$ in the Euler method with the problem below.
I am trying to understand the page 672 on this book here. The book provides the formula
$\left|y(x_{n})-y_{n}\right|\leq \left(\frac{M}{2L}\right)\left(e^{L(x_{n}-x_{0})}-1\right)h$
about the error where you can see the M and L (which looks like some use of Lambert function is needed or rough estimate for the upper bound, look at the $L$ term). There is also an example where it finds some upper bounds and claims some rough estimate. More precisely, I am trying to apply the method of deducing the error term on pages 673-674 for the problem 2 on page 676.
M
I cannot yet understand why the second derivative is used as an estimate for the $M$. On page 673, it just claims that assume that |y''(x)|\leq M but cannot find any premise for it, the 2
in the above. This point about M on pages 673-674 is something black magic to me. Please, explain.
L
The L is apparently just length of the interval.