This help page gives a few examples of using the command ODESteps to solve systems of ordinary differential equations.

See Student[ODEs][ODESteps] for a general description of the command ODESteps and its calling sequence.

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

$\mathrm{high\_order\_ode1}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)+3\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+4\mathrm{diff}\left(y\left(x\right)\,x\right)+2y\left(x\right)=0$

$\mathrm{high\_order\_ode1}≔\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)+3\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+4\frac{\ⅆ}{\ⅆx}y\left(x\right)+2y\left(x\right)=0$

$\mathrm{ODESteps}\left(\mathrm{high\_order\_ode1}\right)$

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

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

$\mathrm{sys2}≔\mathrm{diff}\left(Y\,x\right)=\mathrm{`\%.`}\left(\mathrm{Matrix}\left(\left[\left[7\,1\right]\,\left[\mathrm{`-`}\left(4\right)\,3\right]\right]\right)\,Y\right)$

$\mathrm{sys2}≔\left[\begin{array}{c}\frac{\ⅆ}{\ⅆx}{y}_1\left(x\right)\\ \frac{\ⅆ}{\ⅆx}{y}_2\left(x\right)\end{array}\right]=\left[\begin{array}{cc}7& 1\\ \mathrm{-4}& 3\end{array}\right]·\left[\begin{array}{c}{y}_1\left(x\right)\\ y_2\left(x\right)\end{array}\right]$

$\mathrm{ODESteps}\left(\mathrm{sys2}\right)$

$\mathrm{sys3}≔\mathrm{diff}\left(Y\,x\right)=\mathrm{Matrix}\left(\left[\left[1\,2\right]\,\left[3\,2\right]\right]\right)·Y+⟨1\,\exp \left(x\right)⟩$

$\mathrm{sys3}≔\left[\begin{array}{c}\frac{\ⅆ}{\ⅆx}{y}_1\left(x\right)\\ \frac{\ⅆ}{\ⅆx}{y}_2\left(x\right)\end{array}\right]=\left[\begin{array}{c}{y}_1\left(x\right)+2y_2\left(x\right)+1\\ 3y_1\left(x\right)+2y_2\left(x\right)+{\ⅇ}^x\end{array}\right]$

$\mathrm{ODESteps}\left(\mathrm{sys3}\right)$

$\mathrm{sys4}≔\left\{\mathrm{diff}\left(w\left(x\right)\,x\right)=w\left(x\right)+2z\left(x\right)\,\mathrm{diff}\left(z\left(x\right)\,x\right)=3w\left(x\right)+2z\left(x\right)+\exp \left(x\right)\right\}$

$\mathrm{sys4}≔\left\{\frac{\ⅆ}{\ⅆx}w\left(x\right)=w\left(x\right)+2z\left(x\right)\,\frac{\ⅆ}{\ⅆx}z\left(x\right)=3w\left(x\right)+2z\left(x\right)+{\ⅇ}^x\right\}$

$\mathrm{ODESteps}\left(\mathrm{sys4}\right)$