problema cauchy metodo iterativo

[Tes]

Buongiorno a tutti!
ho un grandissimo problema!!!
non so proprio come risolvere Problemy di Cauchy.
Ne riporto qui uno...potreste aiutarmi a capire quali sono le cose da fare?! non chiedo calcoli..

PC y'= f(x,y) = \( \displaystyle {\left(\frac{{y}}{{{x}+{1}}}+{x}+{1}\lt{b}\frac{{r}}{\gt}{c}{o}{n}{c}{o}{n}{d}{i}{z}{i}{o}\ne\in{i}{z}{i}{a}\le{y}{\left({0}\right)}={0}\lt{b}\frac{{r}}{\gt}{V}{e}{r}{\quad\text{if}\quad}{i}{c}{a}{r}{e}{c}{h}{e}{l}{a}{s}{o}{l}{u}{z}{i}{o}\neè{d}{a}{t}{a}{d}{a}{y}{\left({x}\right)}=\right.} \)x^2\( \displaystyle +{x}\lt{b}\frac{{r}}{\gt}{V}{e}{r}{\quad\text{if}\quad}{i}{c}{a}{r}{e}\in{o}\lt{r}{e}{s}{e}{i}{l}{m}{e}\to{d}{o}\nu{m}{e}{r}{i}{c}{o}{d}{a}\to{d}{a}\lt{b}\frac{{r}}{\gt} \) u_(i+1) \( \displaystyle = \)u_i \( \displaystyle +{h} \)\varphi\( \displaystyle {\left(\right.} \)x_i\( \displaystyle , \)u_i\( \displaystyle , \) x_(i+1) \( \displaystyle , \)u_(i+1) \( \displaystyle :{h}\)\lt{b}\frac{{r}}{\gt}{c}{o}{n} \)\varphi\( \displaystyle {\left(\right.} \)x_i\( \displaystyle , \)u_i\( \displaystyle , \) x_(i+1) \( \displaystyle , \)u_(i+1) \( \displaystyle ,{h}\)+\frac{{1}}{{6}}{f{{\left(\right.}}} \)x_(i+1) \( \displaystyle , \)u_(i+1) \( \displaystyle \)+\frac{{5}}{{6}}{f{{\left(\right.}}} \)x_i\( \displaystyle , \)u_i\( \displaystyle \)\lt{b}\frac{{r}}{\gt}{s}{i}{a}{c}{o}{n}{s}{i}{s}{t}{e}{n}{t}{e}{e}{s}{e},{p}{e}{r}{h}=\frac{{1}}{{2}}{e}{x} \)in\( \displaystyle {\left[{0},{1}\right]}{i}{l}{m}{e}\to{d}{o}{i}{t}{e}{r}{a}{t}{i}{v}{o}{u}{t}{i}{l}{i}{z}{z}{a}\to{p}{e}{r}{r}{i}{s}{o}{l}{v}{e}{r}{e}{l}'{e}{q}{u}{a}{z}{i}{o}\ne\in{f{{\quad\text{or}\quad}}}{m}{a}{i}{m}{p}{l}{i}{c}{i}{t}{a}{s}{i}{a}{c}{o}{n}{v}{e}{r}\ge{n}{t}{e}.\lt{b}\frac{{r}}{\gt}{C}{a}{l}{c}{o}{l}{a}{r}{e}{i}{p}{r}{i}{m}{d}{u}{e}{p}{a}{s}{s}{i}\partial{m}{e}\to{d}{o}{e}\int{e}{r}{p}{o}{l}{a}{r}{e}{i}{r}{i}{s}{\underline{{t}}}{a}{t}{i}{c}{o}{n}{u}{n}{p}{o}{l}\in{o}{m}{i}{o}{d}{i}\nabla{o}\min{\quad\text{or}\quad}{e}{o}{u}{g{{u}}}{a}\le{a}{2}.\lt{b}\frac{{r}}{\gt}{V}{e}{r}{\quad\text{if}\quad}{i}{c}{a}{r}{e}\in{o}\lt{r}{e}{l}'{e}{r}{r}{\quad\text{or}\quad}{e}{c}{o}{m}{m}{e}{s}{s}{o}\ne{l}{l}'\int{e}{r}{v}{a}{l}{l}{o}{\left[{0},{1}\right]}\lt{b}\frac{{r}}{\gt}\lt{b}\frac{{r}}{\gt}\lt{b}\frac{{r}}{\gt}{p}{e}{r}{v}{e}{r}{\quad\text{if}\quad}{i}{c}{a}{r}{e}{l}{a}{c}{o}{n}{s}{i}{s}{t}{e}{n}{z}{a}{d}{e}{v}{o}{v}{e}{r}{\quad\text{if}\quad}{i}{c}{a}{r}{e}{s}{e} \)\alpha\( \displaystyle + \)\beta$ = 1 , cioè 1/6 + 5/6 = 1
per la convergenza non so proprio come fare...


vi ringrazio!

[nato_pigro]

vedo scritto male solo io? :S