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.

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

$\mathrm{high\_order\_ivp1}≔\left\{\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)+3\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+4\mathrm{diff}\left(y\left(x\right)\,x\right)+2y\left(x\right)=0\,\mathrm{eval}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\,x=0\right)=-1\,\mathrm{eval}\left(\mathrm{diff}\left(y\left(x\right)\,x\,x\right)\,x=0\right)=2\,y\left(0\right)=1\right\}$

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

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

$\begin{array}{lll}& & \text{Let's solve}\\ & & \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\}\\ \text{•}& & \text{Highest derivative means the order of the ODE is}3\\ & & \frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\\ \text{•}& & \text{Characteristic polynomial of ODE}\\ & & r^3+3r^2+4r+2=0\\ \text{•}& & \text{Roots of the characteristic polynomial}\\ & & r=\left[\mathrm{-1}\,\mathrm{-1}-\mathrm{I}\,\mathrm{-1}+\mathrm{I}\right]\\ \text{•}& & \text{Solution from}r=\mathrm{-1}\\ & & y_1\left(x\right)={\ⅇ}^{-x}\\ \text{•}& & \text{Solutions from}r=\mathrm{-1}-\mathrm{I}\text{and}r=\mathrm{-1}+\mathrm{I}\\ & & \left[y_2\left(x\right)={\ⅇ}^{-x}\sin \left(x\right)\,y_3\left(x\right)={\ⅇ}^{-x}\cos \left(x\right)\right]\\ \text{•}& & \text{General solution of the ODE}\\ & & y\left(x\right)=\mathrm{c\_\_1}{y}_1\left(x\right)+\mathrm{c\_\_2}{y}_2\left(x\right)+\mathrm{c\_\_3}{y}_3\left(x\right)\\ \text{•}& & \text{Substitute in solutions and simplify}\\ & & y\left(x\right)={\ⅇ}^{-x}\left(\mathrm{c\_\_1}+\mathrm{c\_\_2}\sin \left(x\right)+\mathrm{c\_\_3}\cos \left(x\right)\right)\\ \text{•}& & \text{Use the initial condition}y\left(0\right)=1\\ & & 1=\mathrm{c\_\_1}+\mathrm{c\_\_3}\\ \text{•}& & \text{Calculate the 1st derivative of the solution}\\ & & \frac{\ⅆ}{\ⅆx}y\left(x\right)=-{\ⅇ}^{-x}\left(\mathrm{c\_\_1}+\mathrm{c\_\_2}\sin \left(x\right)+\mathrm{c\_\_3}\cos \left(x\right)\right)+{\ⅇ}^{-x}\left(\mathrm{c\_\_2}\cos \left(x\right)-\mathrm{c\_\_3}\sin \left(x\right)\right)\\ \text{•}& & \text{Use the initial condition}\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}\\ & & \mathrm{-1}=-\mathrm{c\_\_1}-\mathrm{c\_\_3}+\mathrm{c\_\_2}\\ \text{•}& & \text{Calculate the 2nd derivative of the solution}\\ & & \frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)={\ⅇ}^{-x}\left(\mathrm{c\_\_1}+\mathrm{c\_\_2}\sin \left(x\right)+\mathrm{c\_\_3}\cos \left(x\right)\right)-2{\ⅇ}^{-x}\left(\mathrm{c\_\_2}\cos \left(x\right)-\mathrm{c\_\_3}\sin \left(x\right)\right)+{\ⅇ}^{-x}\left(-\mathrm{c\_\_2}\sin \left(x\right)-\mathrm{c\_\_3}\cos \left(x\right)\right)\\ \text{•}& & \text{Use the initial condition}\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\\ & & 2=\mathrm{c\_\_1}-2\mathrm{c\_\_2}\\ \text{•}& & \text{Solve for the unknown coefficients}\\ & & \left\{\mathrm{c\_\_1}=2\,\mathrm{c\_\_2}=0\,\mathrm{c\_\_3}=\mathrm{-1}\right\}\\ \text{•}& & \text{Solution to the IVP}\\ & & y\left(x\right)={\ⅇ}^{-x}\left(2-\cos \left(x\right)\right)\end{array}$

$\mathrm{macro}\left(Y=⟨y\left[1\right]\left(x\right)\,y\left[2\right]\left(x\right)⟩\right)\:$

$\mathrm{ivpsys2}≔\left\{\mathrm{diff}\left(Y\,x\right)=\mathrm{`\%.`}\left(\mathrm{Matrix}\left(\left[\left[7\,1\right]\,\left[\mathrm{`-`}\left(4\right)\,3\right]\right]\right)\,Y\right)\,\mathrm{eval}\left(Y\,x=0\right)=⟨1\,1⟩\right\}$

$\mathrm{ivpsys2}≔\left\{\left[\begin{array}{c}\frac{\ⅆ}{\ⅆx}{y}_1\left(x\right)\\ \frac{\ⅆ}{\ⅆx}{y}_2\left(x\right)\end{array}\right]=\left[\begin{array}{cc}7& 1\\ \mathrm{-4}& 3\end{array}\right]·\left[\begin{array}{c}{y}_1\left(x\right)\\ y_2\left(x\right)\end{array}\right]\,\left[\begin{array}{c}{y}_1\left(0\right)\\ y_2\left(0\right)\end{array}\right]=\left[\begin{array}{c}1\\ 1\end{array}\right]\right\}$

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

$\mathrm{ivpsys3}≔\left\{\mathrm{diff}\left(Y\,x\right)=\mathrm{Matrix}\left(\left[\left[1\,2\right]\,\left[3\,2\right]\right]\right)·Y+⟨1\,\exp \left(x\right)⟩\,\mathrm{eval}\left(Y\,x=1\right)=⟨0\,-1⟩\right\}$

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

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

$\mathrm{ivpsys4}≔\left\{\mathrm{diff}\left(w\left(x\right)\,x\right)=w\left(x\right)+2z\left(x\right)\,\mathrm{diff}\left(z\left(x\right)\,x\right)=3w\left(x\right)+2z\left(x\right)+\exp \left(x\right)\,w\left(-1\right)=2\,z\left(-1\right)=-2\right\}$

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

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