Solving an ODE using Picards Iteration technique

Question

Please note I have solved parts a and b, I have also calculated the value of $L$ using the formula given, however I don't know how to calculate for $M$. It says $\max|f(t,y)|$ so am I supposed to use $y(t)=y_o+f(t_o,y_o)\,(t-t_o)$? I used that but then $y(t)$ will be $0$ and therefore $\max|f(t,y)|$ will become only $t^2$. The value of $t$ which I got while calculating $L$ is $1.03393$ and using that $h<1/L$ condition is satisfied, however $Mh<b$ condition is not being satisfied.

Can someone help me on this? Any help will be greatly appreciated.

A similar task (but for a different ODE) of determining a rectangle for the Picard fixed-point iteration was discussed in The IVP $\dot{x}=x^3+e^{-t^2}$, $x(0)=1$ possesses a solution in $I=(-1/9,1/9)$ — Lutz Lehmann, Nov 29 '19 at 19:28

score 0 · Accepted Answer · answered Nov 29 '19 at 19:14

The formulation of part b) is somewhat ambiguous. What is meant is that the function $f$ is defined as $f(t,y)=t^2+y^2$, and then the differential equation, the classical quadratic Riccati equation, is $y'(t)=f(t,y(t))=t^2+y(t)^2$. The first is an ordinary function definition, the second a functional equation for $y(t)$.

Thus the easiest part is to determine the bounds, $$ M=\max_{|t|\le h,\,|y|\le b} |f(t,y)|=h^2+b^2, $$ and the Lipschitz constant is determined from $$ |f(t,y_1)-f(t,y_2)|\le 2b|y_1-y_2| $$ as $L=2b$.

Now you have to balance the equations $$ q=Lh=2bh<1~\text{ and }~ Mh=(b^2+h^2)h\le b. $$ (Or use strict inequalities everywhere). Or rather confirm that indeed with $h=\frac12$ you get $q=b=\frac17<1$ and $Mh=\frac1{98}+\frac18<\frac17\iff 98>56$.

Thanks a lot sir, I got it now, :) – Math_Is_Fun Nov 30 '19 at 02:35 — Math_Is_Fun, Nov 30 '19 at 02:35

Solving an ODE using Picards Iteration technique

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