The matrixDE command solves a system of linear ODEs of the form $X\'\left(t\right)=A\left(t\right)X\left(t\right)+B\left(t\right)$. If B is not specified then it is assumed to be the zero vector.

An option of the form method = matrixexp can be specified to use matrix exponentials (in the case of constant coefficients).

An option of the form solution = polynomial or solution = rational can be specified to search for polynomial or rational solution. In this case, the function invokes LinearFunctionalSystems[PolynomialSolution] or LinearFunctionalSystems[RationalSolution].

The command returns a pair $\left[S\left(t\right)\,P\left(t\right)\right]$ with $S\left(t\right)$, which is an $n$ by $n$ matrix, and $P\left(t\right)$, which is an $n$ by $1$ vector. If you want the result expressed using Matrix instead, then use convert/Matrix on S(t).

A particular solution of the system can be then written in the form $F\left(t\right)=S\left(t\right)\mathrm{C0}+P\left(t\right)$ where $\mathrm{C0}$ is $n$ by $1$ and $F\left(0\right)=\mathrm{C0}+P\left(0\right)$. If B is zero then P will also be zero.

If a system is expressed in terms of equations, dsolve can be used instead.

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

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

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

$\mathrm{sol}≔\mathrm{matrixDE}\left(A\,t\right)$

$\mathrm{sol}≔\left[\left[\begin{array}{cc}{\ⅇ}^t{t}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{BesselI}\left(\frac{3}{5}\,\frac{2t^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{5}\right)& {\ⅇ}^t{t}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{BesselK}\left(\frac{3}{5}\,\frac{2t^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{5}\right)\\ {\ⅇ}^t\mathrm{BesselI}\left(-\frac{2}{5}\,\frac{2t^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{5}\right)t& -{\ⅇ}^t\mathrm{BesselK}\left(\frac{2}{5}\,\frac{2t^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{5}\right)t\end{array}\right]\,\left[\begin{array}{cc}0& 0\end{array}\right]\right]$

$C≔\mathrm{Matrix}\left(2\,1\right)\:$

$F≔\mathrm{evalm}\left(\mathrm{sol}\left[1\right]\&*C+\mathrm{sol}\left[2\right]\right)\:$$\mathrm{rh}≔\mathrm{evalm}\left(A\&*F\right)\:$

$\mathrm{simplify}\left(\mathrm{normal}\left(\mathrm{diff}\left(F\left[1,1\right]\,t\right)-\mathrm{rh}\left[1,1\right]\right)\,\mathrm{symbolic}\right)$

$0$

$\mathrm{simplify}\left(\mathrm{normal}\left(\mathrm{diff}\left(F\left[2,1\right]\,t\right)-\mathrm{rh}\left[2,1\right]\right)\,\mathrm{symbolic}\right)$

$0$

$A≔\mathrm{Matrix}\left(2\,2\,\left[1\,1\,0\,1\right]\right)\;$$B≔\mathrm{Matrix}\left(2\,1\,\left[t^k\,t^l\right]\right)$

$A≔\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$

$B≔\left[\begin{array}{c}{t}^k\\ t^l\end{array}\right]$

$\mathrm{sol}≔\mathrm{matrixDE}\left(A\,B\,t\right)$

$\mathrm{sol}≔\left[\left[\begin{array}{cc}{\ⅇ}^t& {\ⅇ}^{t}t\\ 0& {\ⅇ}^t\end{array}\right]\,\left[\begin{array}{cc}\frac{-{\ⅇ}^{\frac{t}{2}}{t}^{\frac{l}{2}}\mathrm{WhittakerM}\left(\frac{l}{2}\,\frac{l}{2}+\frac{1}{2}\,t\right)kl+k{t}^{\frac{l}{2}+1}\mathrm{WhittakerM}\left(\frac{l}{2}\,\frac{l}{2}+\frac{1}{2}\,t\right){\ⅇ}^{\frac{t}{2}}+{\ⅇ}^{\frac{t}{2}}{t}^{\frac{k}{2}}\mathrm{WhittakerM}\left(\frac{k}{2}\,\frac{k}{2}+\frac{1}{2}\,t\right)l-{\ⅇ}^{\frac{t}{2}}{t}^{\frac{l}{2}}\mathrm{WhittakerM}\left(\frac{l}{2}\,\frac{l}{2}+\frac{1}{2}\,t\right)k-{\ⅇ}^{\frac{t}{2}}{t}^{\frac{l}{2}}\mathrm{WhittakerM}\left(\frac{l}{2}\,\frac{l}{2}+\frac{1}{2}\,t\right)l+t^{l+1}kl+t^{\frac{l}{2}+1}\mathrm{WhittakerM}\left(\frac{l}{2}\,\frac{l}{2}+\frac{1}{2}\,t\right){\ⅇ}^{\frac{t}{2}}+{\ⅇ}^{\frac{t}{2}}{t}^{\frac{k}{2}}\mathrm{WhittakerM}\left(\frac{k}{2}\,\frac{k}{2}+\frac{1}{2}\,t\right)-{\ⅇ}^{\frac{t}{2}}{t}^{\frac{l}{2}}\mathrm{WhittakerM}\left(\frac{l}{2}\,\frac{l}{2}+\frac{1}{2}\,t\right)+t^{l+1}k+t^{l+1}l+t^{l+1}}{\left(k+1\right)\left(l+1\right)}& \frac{-{\ⅇ}^{\frac{t}{2}}{t}^{\frac{l}{2}}\mathrm{WhittakerM}\left(\frac{l}{2}+1\,\frac{l}{2}+\frac{1}{2}\,t\right)l^2+t^{\frac{l}{2}+1}{\ⅇ}^{\frac{t}{2}}\mathrm{WhittakerM}\left(\frac{l}{2}+1\,\frac{l}{2}+\frac{1}{2}\,t\right)l+{\ⅇ}^{\frac{t}{2}}{t}^{\frac{k}{2}}\mathrm{WhittakerM}\left(\frac{k}{2}+1\,\frac{k}{2}+\frac{1}{2}\,t\right)l-2{\ⅇ}^{\frac{t}{2}}{t}^{\frac{l}{2}}\mathrm{WhittakerM}\left(\frac{l}{2}+1\,\frac{l}{2}+\frac{1}{2}\,t\right)l+t^{l+1}{l}^2-t^{l+1}lt+t^{\frac{l}{2}+1}\mathrm{WhittakerM}\left(\frac{l}{2}\,\frac{l}{2}+\frac{1}{2}\,t\right){\ⅇ}^{\frac{t}{2}}+t^{\frac{l}{2}+1}{\ⅇ}^{\frac{t}{2}}\mathrm{WhittakerM}\left(\frac{l}{2}+1\,\frac{l}{2}+\frac{1}{2}\,t\right)+{\ⅇ}^{\frac{t}{2}}{t}^{\frac{k}{2}}\mathrm{WhittakerM}\left(\frac{k}{2}+1\,\frac{k}{2}+\frac{1}{2}\,t\right)-{\ⅇ}^{\frac{t}{2}}{t}^{\frac{l}{2}}\mathrm{WhittakerM}\left(\frac{l}{2}+1\,\frac{l}{2}+\frac{1}{2}\,t\right)-t^{k}lt+2t^{l+1}l-t^{l+1}t-t^{k}t+t^{l+1}}{\left(l+1\right)t}\end{array}\right]\right]$

$F≔\mathrm{evalm}\left(\mathrm{sol}\left[1\right]\&*C+\mathrm{sol}\left[2\right]\right)\:$$\mathrm{rh}≔\mathrm{evalm}\left(A\&*F+B\right)\:$

$\mathrm{simplify}\left(\mathrm{normal}\left(\mathrm{diff}\left(F\left[1,1\right]\,t\right)-\mathrm{rh}\left[1,1\right]\right)\,\mathrm{symbolic}\right)$

$0$

$\mathrm{simplify}\left(\mathrm{normal}\left(\mathrm{diff}\left(F\left[2,1\right]\,t\right)-\mathrm{rh}\left[2,1\right]\right)\,\mathrm{symbolic}\right)$

$0$

$A≔\mathrm{Matrix}\left(2\,2\,\left[1\,0\,1\,f\left(t\right)\right]\right)$

$A≔\left[\begin{array}{cc}1& 0\\ 1& f\left(t\right)\end{array}\right]$

$\mathrm{sol}≔\mathrm{matrixDE}\left(A\,t\right)$

$\mathrm{sol}≔\left[\left[\begin{array}{c}-f\left(t\right)\mathrm{DESol}\left(\left\{\frac{{\ⅆ}^2}{\ⅆt^2}\mathrm{\_Y}\left(t\right)-\left(\frac{\ⅆ}{\ⅆt}\mathrm{\_Y}\left(t\right)\right)f\left(t\right)-\frac{\ⅆ}{\ⅆt}\mathrm{\_Y}\left(t\right)-\mathrm{\_Y}\left(t\right)\left(\frac{\ⅆ}{\ⅆt}f\left(t\right)\right)+f\left(t\right)\mathrm{\_Y}\left(t\right)\right\}\,\left\{\mathrm{\_Y}\left(t\right)\right\}\right)+\frac{\ⅆ}{\ⅆt}\mathrm{DESol}\left(\left\{\frac{{\ⅆ}^2}{\ⅆt^2}\mathrm{\_Y}\left(t\right)-\left(\frac{\ⅆ}{\ⅆt}\mathrm{\_Y}\left(t\right)\right)f\left(t\right)-\frac{\ⅆ}{\ⅆt}\mathrm{\_Y}\left(t\right)-\mathrm{\_Y}\left(t\right)\left(\frac{\ⅆ}{\ⅆt}f\left(t\right)\right)+f\left(t\right)\mathrm{\_Y}\left(t\right)\right\}\,\left\{\mathrm{\_Y}\left(t\right)\right\}\right)\\ \mathrm{DESol}\left(\left\{\frac{{\ⅆ}^2}{\ⅆt^2}\mathrm{\_Y}\left(t\right)-\left(\frac{\ⅆ}{\ⅆt}\mathrm{\_Y}\left(t\right)\right)f\left(t\right)-\frac{\ⅆ}{\ⅆt}\mathrm{\_Y}\left(t\right)-\mathrm{\_Y}\left(t\right)\left(\frac{\ⅆ}{\ⅆt}f\left(t\right)\right)+f\left(t\right)\mathrm{\_Y}\left(t\right)\right\}\,\left\{\mathrm{\_Y}\left(t\right)\right\}\right)\end{array}\right]\,\left[\begin{array}{cc}0& 0\end{array}\right]\right]$

$A≔\mathrm{Matrix}\left(2\,2\,\left[a\,b\,c\,d\right]\right)$

$A≔\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$B≔\mathrm{Matrix}\left(2\,1\,\left[f\left(t\right)\,g\left(t\right)\right]\right)$

$B≔\left[\begin{array}{c}f\left(t\right)\\ g\left(t\right)\end{array}\right]$

$\mathrm{sol}≔\mathrm{matrixDE}\left(A\,B\,t\right)$

$\mathrm{sol}≔\left[\left[\begin{array}{cc}{\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}& {\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\\ -\frac{{\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\left(-d+a+\sqrt{a^2-2ad+4bc+d^2}\right)}{2b}& \frac{{\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\left(d-a+\sqrt{a^2-2ad+4bc+d^2}\right)}{2b}\end{array}\right]\,\left[\begin{array}{cc}\frac{\left(\int \left(-f\left(t\right)d+\frac{\ⅆ}{\ⅆt}f\left(t\right)+bg\left(t\right)\right){\ⅇ}^{-\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\ⅆt\right){\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}-\left(\int \left(-f\left(t\right)d+\frac{\ⅆ}{\ⅆt}f\left(t\right)+bg\left(t\right)\right){\ⅇ}^{\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\ⅆt\right){\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}}{\sqrt{a^2-2ad+4bc+d^2}}& -\frac{2{\ⅇ}^{\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}{\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}g\left(t\right)b-2{\ⅇ}^{\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}{\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}f\left(t\right)d-2{\ⅇ}^{-\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}{\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}g\left(t\right)b+2{\ⅇ}^{-\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}{\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}f\left(t\right)d+2{\ⅇ}^{\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}{\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\left(\frac{\ⅆ}{\ⅆt}f\left(t\right)\right)-2{\ⅇ}^{-\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}{\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\left(\frac{\ⅆ}{\ⅆt}f\left(t\right)\right)-\left(\int \left(-f\left(t\right)d+\frac{\ⅆ}{\ⅆt}f\left(t\right)+bg\left(t\right)\right){\ⅇ}^{-\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\ⅆt\right){\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\sqrt{a^2-2ad+4bc+d^2}+\left(\int \left(-f\left(t\right)d+\frac{\ⅆ}{\ⅆt}f\left(t\right)+bg\left(t\right)\right){\ⅇ}^{-\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\ⅆt\right){\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}a-\left(\int \left(-f\left(t\right)d+\frac{\ⅆ}{\ⅆt}f\left(t\right)+bg\left(t\right)\right){\ⅇ}^{-\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\ⅆt\right){\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}d-\left(\int \left(-f\left(t\right)d+\frac{\ⅆ}{\ⅆt}f\left(t\right)+bg\left(t\right)\right){\ⅇ}^{\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\ⅆt\right){\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\sqrt{a^2-2ad+4bc+d^2}-\left(\int \left(-f\left(t\right)d+\frac{\ⅆ}{\ⅆt}f\left(t\right)+bg\left(t\right)\right){\ⅇ}^{\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\ⅆt\right){\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}a+\left(\int \left(-f\left(t\right)d+\frac{\ⅆ}{\ⅆt}f\left(t\right)+bg\left(t\right)\right){\ⅇ}^{\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\ⅆt\right){\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}d+2f\left(t\right)\sqrt{a^2-2ad+4bc+d^2}}{2b\sqrt{a^2-2ad+4bc+d^2}}\end{array}\right]\right]$

$\mathrm{sol}≔\mathrm{matrixDE}\left(A\,t\,\mathrm{method}=\mathrm{matrixexp}\right)$

$\mathrm{sol}≔\left[\left[\begin{array}{cc}\frac{\sqrt{a^2-2ad+4bc+d^2}{\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}+a{\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}-d{\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}+\sqrt{a^2-2ad+4bc+d^2}{\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}-a{\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}+d{\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}}{2\sqrt{a^2-2ad+4bc+d^2}}& \frac{b\left({\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}-{\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\right)}{\sqrt{a^2-2ad+4bc+d^2}}\\ \frac{c\left({\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}-{\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}\right)}{\sqrt{a^2-2ad+4bc+d^2}}& \frac{\sqrt{a^2-2ad+4bc+d^2}{\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}-a{\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}+d{\ⅇ}^{\frac{\left(a+d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}+\sqrt{a^2-2ad+4bc+d^2}{\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}+a{\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}-d{\ⅇ}^{-\frac{\left(-a-d+\sqrt{a^2-2ad+4bc+d^2}\right)t}{2}}}{2\sqrt{a^2-2ad+4bc+d^2}}\end{array}\right]\,\left[\begin{array}{cc}0& 0\end{array}\right]\right]$

$M≔\mathrm{Matrix}\left(6\,6\,\left[0\,1\,0\,0\,0\,0\,0\,0\,1\,1\,0\,0\,0\,0\,0\,0\,1\,0\,2\,2x\,0\,0\,1\,0\,x^2\,0\,2x\,0\,0\,1\,0\,x^2\,-2\,0\,0\,0\right]\right)$

$M≔\left[\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ 0& 0& 1& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 2& 2x& 0& 0& 1& 0\\ x^2& 0& 2x& 0& 0& 1\\ 0& x^2& \mathrm{-2}& 0& 0& 0\end{array}\right]$

$\mathrm{sol}≔\mathrm{matrixDE}\left(M\,x\,\mathrm{solution}=\mathrm{polynomial}\right)$

$\mathrm{sol}≔\left[\left[\begin{array}{cccccc}0& 1& x& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 1& -x& -x^2& 0& 0& 0\\ \mathrm{-1}& x& x^2& 0& 0& 0\\ 0& \mathrm{-1}& -2x& 0& 0& 0\\ -2x& x^2& x^3-2& 0& 0& 0\end{array}\right]\,\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\end{array}\right]\right]$