Systems of ODEs with IVP - Maple Help

ODE Steps for Systems of ODEs with IVP

Overview

 • This help page gives a few examples of using the command ODESteps to solve systems of ordinary differential equations with initial values.
 • See Student[ODEs][ODESteps] for a general description of the command ODESteps and its calling sequence.

Examples

 > $\mathrm{with}\left(\mathrm{Student}:-\mathrm{ODEs}\right):$
 > $\mathrm{high_order_ivp1}≔\left\{\frac{{ⅆ}^{3}}{ⅆ{x}^{3}}y\left(x\right)+3\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)+4\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+2y\left(x\right)=0,\genfrac{}{}{0}{}{\frac{ⅆ}{ⅆx}y\left(x\right)}{\phantom{x=0}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{ⅆ}{ⅆx}y\left(x\right)}}{x=0}=-1,\genfrac{}{}{0}{}{\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)}{\phantom{x=0}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)}}{x=0}=2,y\left(0\right)=1\right\}$
 ${\mathrm{high_order_ivp1}}{≔}\left\{\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{3}{}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{4}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{2}{}{y}{}\left({x}\right){=}{0}{,}\genfrac{}{}{0}{}{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}}{\left\{{x}{=}{0}\right\}}{=}{2}{,}\genfrac{}{}{0}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{\phantom{\left\{{x}{=}{0}\right\}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}}{\left\{{x}{=}{0}\right\}}{=}{-1}{,}{y}{}\left({0}\right){=}{1}\right\}$ (1)
 > $\mathrm{ODESteps}\left(\mathrm{high_order_ivp1}\right)$
 > $\mathrm{macro}\left(Y=⟨{y}_{1}\left(x\right),{y}_{2}\left(x\right)⟩\right):$
 > $\mathrm{ivpsys2}≔\left\{\frac{\partial }{\partial x}Y=\mathrm{%.}\left(\mathrm{Matrix}\left(\left[\left[7,1\right],\left[-4,3\right]\right]\right),Y\right),\genfrac{}{}{0}{}{Y}{\phantom{x=0}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{Y}}{x=0}=⟨1,1⟩\right\}$
 ${\mathrm{ivpsys2}}{≔}\left\{\left[\begin{array}{c}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{1}}{}\left({x}\right)\\ \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{2}}{}\left({x}\right)\end{array}\right]{=}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%.}}{}\left({\mathrm{RTABLE}}{}\left({36893628113800856988}{,}\left[\begin{array}{cc}{7}& {1}\\ {-4}& {3}\end{array}\right]{,}{\mathrm{Matrix}}\right){,}\left[\begin{array}{c}{y}_{1}{}\left(x\right)\\ {y}_{2}{}\left(x\right)\end{array}\right]\right)\right]\right){,}\left[\begin{array}{c}{{y}}_{{1}}{}\left({0}\right)\\ {{y}}_{{2}}{}\left({0}\right)\end{array}\right]{=}\left[\begin{array}{c}{1}\\ {1}\end{array}\right]\right\}$ (2)
 > $\mathrm{ODESteps}\left(\mathrm{ivpsys2}\right)$
 > $\mathrm{ivpsys3}≔\left\{\frac{\partial }{\partial x}Y=\mathrm{.}\left(\mathrm{Matrix}\left(\left[\left[1,2\right],\left[3,2\right]\right]\right),Y\right)+⟨1,{ⅇ}^{x}⟩,\genfrac{}{}{0}{}{Y}{\phantom{x=1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{Y}}{x=1}=⟨0,-1⟩\right\}$
 ${\mathrm{ivpsys3}}{≔}\left\{\left[\begin{array}{c}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{1}}{}\left({x}\right)\\ \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{2}}{}\left({x}\right)\end{array}\right]{=}\left[\begin{array}{c}{{y}}_{{1}}{}\left({x}\right){+}{2}{}{{y}}_{{2}}{}\left({x}\right){+}{1}\\ {3}{}{{y}}_{{1}}{}\left({x}\right){+}{2}{}{{y}}_{{2}}{}\left({x}\right){+}{{ⅇ}}^{{x}}\end{array}\right]{,}\left[\begin{array}{c}{{y}}_{{1}}{}\left({1}\right)\\ {{y}}_{{2}}{}\left({1}\right)\end{array}\right]{=}\left[\begin{array}{c}{0}\\ {-1}\end{array}\right]\right\}$ (3)
 > $\mathrm{ODESteps}\left(\mathrm{ivpsys3}\right)$
 > $\mathrm{ivpsys4}≔\left\{\frac{ⅆ}{ⅆx}w\left(x\right)=w\left(x\right)+2z\left(x\right),\frac{ⅆ}{ⅆx}z\left(x\right)=3w\left(x\right)+2z\left(x\right)+{ⅇ}^{x},w\left(-1\right)=2,z\left(-1\right)=-2\right\}$
 ${\mathrm{ivpsys4}}{≔}\left\{\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{w}{}\left({x}\right){=}{w}{}\left({x}\right){+}{2}{}{z}{}\left({x}\right){,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}\right){=}{3}{}{w}{}\left({x}\right){+}{2}{}{z}{}\left({x}\right){+}{{ⅇ}}^{{x}}{,}{w}{}\left({-1}\right){=}{2}{,}{z}{}\left({-1}\right){=}{-2}\right\}$ (4)
 > $\mathrm{ODESteps}\left(\mathrm{ivpsys4}\right)$