The System(SYS, Y) command finds the solution of a system of first order linear ODEs.

The system SYS may be written as a list or set of ODEs. If the solving variables cannot be unambiguously determined from the form of SYS, Y must also be specified as a list or set containing the solving variables.

Alternatively, SYS may be written as a single equation of the form:

where Y is a Vector of solving variables, DY a Vector of their derivatives, A is the Matrix of coefficients, and F is the Vector of forcing functions. In this case, Y does not need to be specified as an extra argument since it can be determined from the form of SYS.

A third syntax, System(A, F, Y), is also available as a shortcut to the above syntax System(DY = A . Y + F).

Use the option output=steps to make this command return an annotated step-by-step solution.  Further control over the format and display of the step-by-step solution is available using the options described in Student:-Basics:-OutputStepsRecord.  The options supported by that command can be passed to this one.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\left[\mathrm{Solve}\right]\right)\:$

$\mathrm{sys1}≔\left\{\mathrm{diff}\left(y\left[1\right]\left(x\right)\,x\right)=7y\left[1\right]\left(x\right)+y\left[2\right]\left(x\right)\,\mathrm{diff}\left(y\left[2\right]\left(x\right)\,x\right)=-4y\left[1\right]\left(x\right)+3y\left[2\right]\left(x\right)\right\}$

$\mathrm{sys1}≔\left\{\frac{\ⅆ}{\ⅆx}{y}_1\left(x\right)=7y_1\left(x\right)+y_2\left(x\right)\,\frac{\ⅆ}{\ⅆx}{y}_2\left(x\right)=-4y_1\left(x\right)+3y_2\left(x\right)\right\}$

$\mathrm{System}\left(\mathrm{sys1}\,\left\{y\left[1\right]\left(x\right)\,y\left[2\right]\left(x\right)\right\}\right)$

$\left\{y_1\left(x\right)=-\frac{{\ⅇ}^{5x}\left(2\mathrm{\_C2}x+2\mathrm{\_C1}+\mathrm{\_C2}\right)}{4}\,y_2\left(x\right)={\ⅇ}^{5x}\left(\mathrm{\_C2}x+\mathrm{\_C1}\right)\right\}$

$\mathrm{sys2}≔\left[\mathrm{diff}\left(y\left[1\right]\left(x\right)\,x\right)=7y\left[1\right]\left(x\right)+y\left[2\right]\left(x\right)+1\,\mathrm{diff}\left(y\left[2\right]\left(x\right)\,x\right)=-4y\left[1\right]\left(x\right)+3y\left[2\right]\left(x\right)+\exp \left(x\right)\right]$

$\mathrm{sys2}≔\left[\frac{\ⅆ}{\ⅆx}{y}_1\left(x\right)=7y_1\left(x\right)+y_2\left(x\right)+1\,\frac{\ⅆ}{\ⅆx}{y}_2\left(x\right)=-4y_1\left(x\right)+3y_2\left(x\right)+{\ⅇ}^x\right]$

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

$\left\{y_1\left(x\right)=\frac{\left(\left(-200x-100\right)\mathrm{\_C2}+260x-200\mathrm{\_C1}+23\right){\ⅇ}^{5x}}{400}+\frac{{\ⅇ}^x}{16}-\frac{3}{25}\,y_2\left(x\right)=\frac{\left(\left(200\mathrm{\_C2}-260\right)x+200\mathrm{\_C1}+107\right){\ⅇ}^{5x}}{200}-\frac{3{\ⅇ}^x}{8}-\frac{4}{25}\right\}$

$\mathrm{sys3}≔\left\{\mathrm{diff}\left(y\left[1\right]\left(x\right)\,x\right)=6y\left[1\right]\left(x\right)-3y\left[2\right]\left(x\right)+1\,\mathrm{diff}\left(y\left[2\right]\left(x\right)\,x\right)=-4y\left[1\right]\left(x\right)+9y\left[2\right]\left(x\right)+\cos \left(x\right)\right\}$

$\mathrm{sys3}≔\left\{\frac{\ⅆ}{\ⅆx}{y}_1\left(x\right)=6y_1\left(x\right)-3y_2\left(x\right)+1\,\frac{\ⅆ}{\ⅆx}{y}_2\left(x\right)=-4y_1\left(x\right)+9y_2\left(x\right)+\cos \left(x\right)\right\}$

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

$\left\{y_1\left(x\right)=\frac{\left(\left(380247\mathrm{\_C1}+59492\right)\sqrt{57}+1140741\mathrm{\_C1}+424080\right){\ⅇ}^{-\frac{\left(-15+\sqrt{57}\right)x}{2}}}{3041976}+\frac{\left(\left(-380247\mathrm{\_C2}-59492\right)\sqrt{57}+1140741\mathrm{\_C2}+424080\right){\ⅇ}^{\frac{\left(15+\sqrt{57}\right)x}{2}}}{3041976}-\frac{123\cos \left(x\right)}{1906}+\frac{45\sin \left(x\right)}{1906}-\frac{3}{14}\,y_2\left(x\right)=\frac{\left(1520988\mathrm{\_C1}+20467\sqrt{57}+176567\right){\ⅇ}^{-\frac{\left(-15+\sqrt{57}\right)x}{2}}}{1520988}+\frac{\left(1520988\mathrm{\_C2}-20467\sqrt{57}+176567\right){\ⅇ}^{\frac{\left(15+\sqrt{57}\right)x}{2}}}{1520988}-\frac{261\cos \left(x\right)}{1906}+\frac{49\sin \left(x\right)}{1906}-\frac{2}{21}\right\}$

$Y≔⟨v\left(x\right)\,w\left(x\right)⟩$

$Y≔\left[\begin{array}{c}v\left(x\right)\\ w\left(x\right)\end{array}\right]$

$A≔⟨⟨7|1⟩\,⟨-4|3⟩⟩$

$A≔\left[\begin{array}{cc}7& 1\\ \mathrm{-4}& 3\end{array}\right]$

$F≔⟨1\,\exp \left(x\right)⟩$

$F≔\left[\begin{array}{c}1\\ {\ⅇ}^x\end{array}\right]$

$\mathrm{sys4}≔\mathrm{diff}\left(Y\,x\right)=A·Y$

$\mathrm{sys4}≔\left[\begin{array}{c}\frac{\ⅆ}{\ⅆx}v\left(x\right)\\ \frac{\ⅆ}{\ⅆx}w\left(x\right)\end{array}\right]=\left[\begin{array}{c}7v\left(x\right)+w\left(x\right)\\ -4v\left(x\right)+3w\left(x\right)\end{array}\right]$

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

$\left\{v\left(x\right)=-\frac{{\ⅇ}^{5x}\left(2\mathrm{\_C2}x+2\mathrm{\_C1}+\mathrm{\_C2}\right)}{4}\,w\left(x\right)={\ⅇ}^{5x}\left(\mathrm{\_C2}x+\mathrm{\_C1}\right)\right\}$

$\mathrm{sys5}≔\mathrm{diff}\left(Y\,x\right)=\mathrm{`\%.`}\left(A\,Y\right)+F$

$\mathrm{sys5}≔\left[\begin{array}{c}\frac{\ⅆ}{\ⅆx}v\left(x\right)\\ \frac{\ⅆ}{\ⅆx}w\left(x\right)\end{array}\right]=\left[\begin{array}{cc}7& 1\\ \mathrm{-4}& 3\end{array}\right]·\left[\begin{array}{c}v\left(x\right)\\ w\left(x\right)\end{array}\right]+\left[\begin{array}{c}1\\ {\ⅇ}^x\end{array}\right]$

$\mathrm{System}\left(\mathrm{sys5}\right)$

$\left\{v\left(x\right)=\frac{\left(\left(-200x-100\right)\mathrm{\_C2}+260x-200\mathrm{\_C1}+23\right){\ⅇ}^{5x}}{400}+\frac{{\ⅇ}^x}{16}-\frac{3}{25}\,w\left(x\right)=\frac{\left(\left(200\mathrm{\_C2}-260\right)x+200\mathrm{\_C1}+107\right){\ⅇ}^{5x}}{200}-\frac{3{\ⅇ}^x}{8}-\frac{4}{25}\right\}$

$B≔⟨⟨1|2⟩\,⟨3|2⟩⟩$

$B≔\left[\begin{array}{cc}1& 2\\ 3& 2\end{array}\right]$

$\mathrm{sys6}≔\mathrm{diff}\left(Y\,x\right)=B·Y+F$

$\mathrm{sys6}≔\left[\begin{array}{c}\frac{\ⅆ}{\ⅆx}v\left(x\right)\\ \frac{\ⅆ}{\ⅆx}w\left(x\right)\end{array}\right]=\left[\begin{array}{c}v\left(x\right)+2w\left(x\right)+1\\ 3v\left(x\right)+2w\left(x\right)+{\ⅇ}^x\end{array}\right]$

$\mathrm{System}\left(B\,F\,Y\right)$

$\left\{v\left(x\right)=\frac{\left(-30\mathrm{\_C1}-12\right){\ⅇ}^{-x}}{30}+\frac{\left(20\mathrm{\_C2}+7\right){\ⅇ}^{4x}}{30}-\frac{{\ⅇ}^x}{3}+\frac{1}{2}\,w\left(x\right)=\frac{\left(20\mathrm{\_C1}+8\right){\ⅇ}^{-x}}{20}-\frac{3}{4}+\frac{\left(20\mathrm{\_C2}+7\right){\ⅇ}^{4x}}{20}\right\}$

The Student[ODEs][Solve][System] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.

The Student[ODEs][Solve][System] command was updated in Maple 2022.

The output option was introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.