Differentialgleichung/Picard-Lindelöf/t^2y-xy^2,x^3-y+t/Beispiel

Wir wenden die Picard-Lindelöf-Iteration auf das Anfangswertproblem

${}v(0)=(0,0)\,$

zum Vektorfeld

$F\colon \mathbb {R} \times \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2},\,(t,x,y)\longmapsto (t^{2}y-xy^{2},x^{3}-y+t)$

an. Es ist

${}\varphi _{0}(t)=(0,0)\,.$

Daher ist

${}\varphi _{1}(t)=\int _{0}^{t}(0,s)ds=\left(0,\,{\frac {1}{2}}t^{2}\right)\,.$

Es ist

${}F(s,\varphi _{1}(s))=\left(s^{2}{\frac {1}{2}}s^{2},\,-{\frac {1}{2}}s^{2}+s\right)=\left({\frac {1}{2}}s^{4},\,-{\frac {1}{2}}s^{2}+s\right)\,$

und daher

${}\varphi _{2}(t)=\int _{0}^{t}\left({\frac {1}{2}}s^{4},\,-{\frac {1}{2}}s^{2}+s\right)ds=\left({\frac {1}{10}}t^{5},\,-{\frac {1}{6}}t^{3}+{\frac {1}{2}}t^{2}\right)\,.$

Wegen

${}{\begin{aligned}F(s,\varphi _{2}(s))&=\left(s^{2}{\left(-{\frac {1}{6}}s^{3}+{\frac {1}{2}}s^{2}\right)}-{\frac {1}{10}}s^{5}{\left(-{\frac {1}{6}}s^{3}+{\frac {1}{2}}s^{2}\right)}^{2},\,{\left({\frac {1}{10}}t^{5}\right)}^{3}-{\left(-{\frac {1}{6}}s^{3}+{\frac {1}{2}}s^{2}\right)}+s\right)\\&=\left(-{\frac {1}{6}}s^{5}+{\frac {1}{2}}s^{4}-{\frac {1}{10}}s^{5}{\left({\frac {1}{36}}s^{6}-{\frac {1}{6}}s^{5}+{\frac {1}{4}}s^{4}\right)},\,{\frac {1}{1000}}s^{15}+{\frac {1}{6}}s^{3}-{\frac {1}{2}}s^{2}+s\right)\\&=\left(-{\frac {1}{360}}s^{11}+{\frac {1}{60}}s^{10}-{\frac {1}{40}}s^{9}-{\frac {1}{6}}s^{5}+{\frac {1}{2}}s^{4},\,{\frac {1}{1000}}s^{15}+{\frac {1}{6}}s^{3}-{\frac {1}{2}}s^{2}+s\right)\end{aligned}}$

und daher

${}{\begin{aligned}\varphi _{3}(t)&=\int _{0}^{t}\left(-{\frac {1}{360}}s^{11}+{\frac {1}{60}}s^{10}-{\frac {1}{40}}s^{9}-{\frac {1}{6}}s^{5}+{\frac {1}{2}}s^{4},\,{\frac {1}{1000}}s^{15}+{\frac {1}{6}}s^{3}-{\frac {1}{2}}s^{2}+s\right)ds\\&=\left(-{\frac {1}{4320}}t^{12}+{\frac {1}{660}}t^{11}-{\frac {1}{400}}t^{10}-{\frac {1}{36}}t^{6}+{\frac {1}{10}}t^{5},\,{\frac {1}{16000}}t^{16}+{\frac {1}{24}}t^{4}-{\frac {1}{6}}t^{3}+{\frac {1}{2}}t^{2}\right).\end{aligned}}$