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.

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{ODEs}\right)\:$

$\mathrm{high\_order\_ivp1}≔\left\{\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\+3\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)\+4\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\+2y\left(x\right)\=0\,\genfrac{}{}{0ex}{}{\frac{\ⅆ}{\ⅆx}y\left(x\right)}{\phantom{x\=0}}|\genfrac{}{}{0ex}{}{\phantom{\frac{\ⅆ}{\ⅆx}y\left(x\right)}}{x\=0}\=-1\,\genfrac{}{}{0ex}{}{\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)}{\phantom{x\=0}}|\genfrac{}{}{0ex}{}{\phantom{\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)}}{x\=0}\=2\,y\left(0\right)\=1\right\}$

$\mathrm{high\_order\_ivp1}≔\left\{\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)+3\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+4\frac{\ⅆ}{\ⅆx}y\left(x\right)+2y\left(x\right)=0\,\genfrac{}{}{0ex}{}{\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}}{\left\{x=0\right\}}=2\,\genfrac{}{}{0ex}{}{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}}{\left\{x=0\right\}}=\mathrm{-1}\,y\left(0\right)=1\right\}$

$\mathrm{ODESteps}\left(\mathrm{high\_order\_ivp1}\right)$

$\mathrm{macro}\left(Y\=⟨y_1\left(x\right)\,y_2\left(x\right)⟩\right)\:$

$\mathrm{ivpsys2}≔\left\{\frac{\partial }{\partial x}Y\=\mathrm{`\%.`}\left(\mathrm{Matrix}\left(\left[\left[7\,1\right]\,\left[-4\,3\right]\right]\right)\,Y\right)\,\genfrac{}{}{0ex}{}{Y}{\phantom{x\=0}}|\genfrac{}{}{0ex}{}{\phantom{Y}}{x\=0}\=⟨1\,1⟩\right\}$

$\mathrm{ivpsys2}≔\left\{\left[\begin{array}{c}\frac{\ⅆ}{\ⅆx}{y}_1\left(x\right)\\ \frac{\ⅆ}{\ⅆx}{y}_2\left(x\right)\end{array}\right]=\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%.}\left(\mathrm{RTABLE}\left(36893628113800856988\,\left[\begin{array}{cc}7& 1\\ \mathrm{-4}& 3\end{array}\right]\,\mathrm{Matrix}\right)\,\left[\begin{array}{c}{y}_1\left(x\right)\\ y_2\left(x\right)\end{array}\right]\right)\right]\right)\,\left[\begin{array}{c}{y}_1\left(0\right)\\ y_2\left(0\right)\end{array}\right]=\left[\begin{array}{c}1\\ 1\end{array}\right]\right\}$

$\mathrm{ODESteps}\left(\mathrm{ivpsys2}\right)$

$\mathrm{ivpsys3}≔\left\{\frac{\partial }{\partial x}Y\=\mathrm{`.`}\left(\mathrm{Matrix}\left(\left[\left[1\,2\right]\,\left[3\,2\right]\right]\right)\,Y\right)\+⟨1\,{\ⅇ}^x⟩\,\genfrac{}{}{0ex}{}{Y}{\phantom{x\=1}}|\genfrac{}{}{0ex}{}{\phantom{Y}}{x\=1}\=⟨0\,-1⟩\right\}$

$\mathrm{ivpsys3}≔\left\{\left[\begin{array}{c}\frac{\ⅆ}{\ⅆx}{y}_1\left(x\right)\\ \frac{\ⅆ}{\ⅆx}{y}_2\left(x\right)\end{array}\right]=\left[\begin{array}{c}{y}_1\left(x\right)+2y_2\left(x\right)+1\\ 3y_1\left(x\right)+2y_2\left(x\right)+{\ⅇ}^x\end{array}\right]\,\left[\begin{array}{c}{y}_1\left(1\right)\\ y_2\left(1\right)\end{array}\right]=\left[\begin{array}{c}0\\ \mathrm{-1}\end{array}\right]\right\}$

$\mathrm{ODESteps}\left(\mathrm{ivpsys3}\right)$

$\mathrm{ivpsys4}≔\left\{\frac{\ⅆ}{\ⅆx}w\left(x\right)\=w\left(x\right)\+2z\left(x\right)\,\frac{\ⅆ}{\ⅆx}z\left(x\right)\=3w\left(x\right)\+2z\left(x\right)\+{\ⅇ}^x\,w\left(-1\right)\=2\,z\left(-1\right)\=-2\right\}$

$\mathrm{ivpsys4}≔\left\{\frac{\ⅆ}{\ⅆx}w\left(x\right)=w\left(x\right)+2z\left(x\right)\,\frac{\ⅆ}{\ⅆx}z\left(x\right)=3w\left(x\right)+2z\left(x\right)+{\ⅇ}^x\,w\left(\mathrm{-1}\right)=2\,z\left(\mathrm{-1}\right)=\mathrm{-2}\right\}$

$\mathrm{ODESteps}\left(\mathrm{ivpsys4}\right)$