The ByVariationOfParameters(SYS, vars) command finds the solution of a system of first order linear ODEs using variation of parameters.

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: ByVariationOfParameters(A, F, x) is also available as a shortcut to the above syntax 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.

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

$A≔\mathrm{Matrix}\left(\left[\left[1\,2\right]\,\left[3\,2\right]\right]\right)$

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

$F≔⟨1\,\exp \left(x\right)⟩$

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

$Y≔⟨\mathrm{y1}\left(x\right)\,\mathrm{y2}\left(x\right)⟩\:$

$\mathrm{sys1}≔\mathrm{diff}\left(Y\,x\right)=A·Y+F$

$\mathrm{sys1}≔\left[\begin{array}{c}\frac{\ⅆ}{\ⅆx}\mathrm{y1}\left(x\right)\\ \frac{\ⅆ}{\ⅆx}\mathrm{y2}\left(x\right)\end{array}\right]=\left[\begin{array}{c}\mathrm{y1}\left(x\right)+2\mathrm{y2}\left(x\right)+1\\ 3\mathrm{y1}\left(x\right)+2\mathrm{y2}\left(x\right)+{\ⅇ}^x\end{array}\right]$

$\mathrm{Student}:-\mathrm{ODEs}:-\mathrm{Solve}:-\mathrm{ByVariationOfParameters}\left(\mathrm{sys1}\right)$

$\left[\begin{array}{c}\mathrm{y1}\left(x\right)\\ \mathrm{y2}\left(x\right)\end{array}\right]=\left[\begin{array}{c}-\frac{{\ⅇ}^x}{3}+\frac{1}{2}-\frac{2{\ⅇ}^{-x}}{5}+\frac{7{\ⅇ}^{4x}}{30}\\ -\frac{3}{4}+\frac{2{\ⅇ}^{-x}}{5}+\frac{7{\ⅇ}^{4x}}{20}\end{array}\right]$

$\mathrm{sys2}≔\left[\mathrm{seq}\left(\mathrm{diff}\left(Y\left[i\right]\,x\right)=\left(A·Y+F\right)\left[i\right]\,i=1..2\right)\right]$

$\mathrm{sys2}≔\left[\frac{\ⅆ}{\ⅆx}\mathrm{y1}\left(x\right)=\mathrm{y1}\left(x\right)+2\mathrm{y2}\left(x\right)+1\,\frac{\ⅆ}{\ⅆx}\mathrm{y2}\left(x\right)=3\mathrm{y1}\left(x\right)+2\mathrm{y2}\left(x\right)+{\ⅇ}^x\right]$

$\mathrm{Student}:-\mathrm{ODEs}:-\mathrm{Solve}:-\mathrm{ByVariationOfParameters}\left(\mathrm{sys2}\right)$

$\left[\mathrm{y1}\left(x\right)=-\frac{{\ⅇ}^x}{3}+\frac{1}{2}-\frac{2{\ⅇ}^{-x}}{5}+\frac{7{\ⅇ}^{4x}}{30}\,\mathrm{y2}\left(x\right)=-\frac{3}{4}+\frac{2{\ⅇ}^{-x}}{5}+\frac{7{\ⅇ}^{4x}}{20}\right]$

$\mathrm{sys3}≔\mathrm{convert}\left(\mathrm{sys2}\,\mathrm{set}\right)$

$\mathrm{sys3}≔\left\{\frac{\ⅆ}{\ⅆx}\mathrm{y1}\left(x\right)=\mathrm{y1}\left(x\right)+2\mathrm{y2}\left(x\right)+1\,\frac{\ⅆ}{\ⅆx}\mathrm{y2}\left(x\right)=3\mathrm{y1}\left(x\right)+2\mathrm{y2}\left(x\right)+{\ⅇ}^x\right\}$

$\mathrm{Student}:-\mathrm{ODEs}:-\mathrm{Solve}:-\mathrm{ByVariationOfParameters}\left(\mathrm{sys3}\right)$

$\left\{\mathrm{y1}\left(x\right)=-\frac{{\ⅇ}^x}{3}+\frac{1}{2}-\frac{2{\ⅇ}^{-x}}{5}+\frac{7{\ⅇ}^{4x}}{30}\,\mathrm{y2}\left(x\right)=-\frac{3}{4}+\frac{2{\ⅇ}^{-x}}{5}+\frac{7{\ⅇ}^{4x}}{20}\right\}$

The Student[ODEs][Solve][ByVariationOfParameters] command was introduced in Maple 2022.

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