Description - Maple Programming Help

Calculate Picard Iterates for the IVP $y\prime =f\left(x,y\right),y\left({x}_{0}\right)={y}_{0}$



 Description Calculate an iterative solution to an ODE by using Picard's method.

Picard Iterates for the IVP $y\prime =f\left(x,y\right),y\left({x}_{0}\right)={y}_{0}$

The function $f\left(x,y\right)$

 > $f≔\left(x,y\right)\to {x}{+}{{y}}^{{2}}$
 ${f}{:=}\left({x}{,}{y}\right){→}{x}{+}{{y}}^{{2}}$ (1)

Set $a={x}_{0}$

 > $a≔{0}$
 ${a}{:=}{0}$ (2)

Set ${\mathrm{φ}}_{0}=y\left({x}_{0}\right)={y}_{0}$

 > ${\mathrm{φ}}_{0}≔{0}$
 ${{\mathrm{φ}}}_{{0}}{:=}{0}$ (3)

Number of iterates

 > $N≔{3}$
 ${N}{:=}{3}$ (4)

Picard Iterates

 >
 ${\mathrm{Iterate number 1:}}$
 $\frac{{1}}{{2}}{}{{x}}^{{2}}$
 ${\mathrm{Iterate number 2:}}$
 $\frac{{1}}{{2}}{}{{x}}^{{2}}{+}\frac{{1}}{{20}}{}{{x}}^{{5}}$
 ${\mathrm{Iterate number 3:}}$
 $\frac{{1}}{{4400}}{}{{x}}^{{11}}{+}\frac{{1}}{{160}}{}{{x}}^{{8}}{+}\frac{{1}}{{20}}{}{{x}}^{{5}}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}$ (5)

 Commands Used