If i have to find interval of unique solution around the point $x_0$ by Picard theorem which is given by $(x_0-h,x_0+h)$ where $h=max~min(a,\frac{b}{M})$ and $M$ is the maximum value of $|f(x,y)|$ over the rectangle $|x-x_0|\leq a, |y-y_0|\leq b$ for the IVP $y'=f(x,y),y(x_0)=y_0$ with all conditions of Picard existence and uniqueness theorem. Now my doubt is about calculation part which is to find $h$. Some authors(like S.L. Ross) find it by assuming $f(b)=\frac{b}{M}$ and find maximum value of $f(b)$ by using elementary calculus so I have no problem in this way. But my friend always avoid this method and he always find $h$ by putting $a=\frac{b}{M}$ and then find some relation in $a$ and $b$ and estimate maximum value of $a$ from there as solved in these examples https://drive.google.com/file/d/16KKGO0jiKOEw-9RCk2fkbkDIg1lzbtUI/view?usp=drivesdk by both ways . Answers in both methods comes same. We(me and my friend) don't know logic behind this second method of putting $a=\frac{b}{M}$. So my questions are
$1.$ is it correct way to find $h$ by putting $a=\frac{b}{M}$ always?
$2.$ if this is correct way then what is logic behind this second method.
Thank you.
Best Answer
The order is always, first one fixes $a,b$, then computes the maximum $M=M(a,b)$ (and confirms the Lipschitz condition), then derives $h=\min(a,\frac{b}{M(a,b)})$ as the interval radius where the solutions can not leave the box/cylinder via its horizontal boundaries.
Now it may happen that the initial length parameters $a,b$ where chosen so fortunate that $h=a=\frac{b}{M(a,b)}$ by this mechanism. If $M(a,b)$ is an easy enough expression, one can help this luck by some computation. As there are two free parameters, one can then optimize $b$ so that $a$ becomes as large as possible.
But none of that is necessary for the theoretical application of this construction in the existence theorem, and usually the results of the above optimization still fall (very) short of other estimates of intervals contained in the maximal domain.
I do not see the difference between the two methods you describe. The first is perhaps a little more formal. Note that if you do this step-by-step, then $M(h,b)$ will most often be smaller than $M(a,b)$, which means that one could relax $h$ a little in direction $a$ and restart with a new $a$ value, etc. Implicitly this is a fixed-point equation for $a$ where the value of $b$ remains unchanged. Having a solution or sufficient approximation of this giving $a=g(b)$ as a function of $b$ with $g(b)\le\frac{b}{M(g(b),b)}$, one can then proceed to find an optimal $b$ as a maximum of $g$.
In your written example calculations, there is no difference between the methods in the cases of the autonomous equations, as $a$ does not influence $M$, so that $g(b)=\frac{b}{M(\cdot,b)}$ directly. In the other cases, I think that the examples are simple enought that the admissible set $\{(a,b):aM(a,b)\le b\}$ is convex so that it does not matter if you approach the boundary in horizontal direction like in method 2 or in vertical direction like in method 1. I do not have counter-examples, but there might be situations where this convexity fails and you get different results from both methods.