The Matrix Riccati differential equation is

with

where $Z,A,K$, and $\mathrm{Z0}$ are n by n matrices. matrix_riccati returns the solution, $Z$.

The command with(DEtools,matrix_riccati) allows the use of the abbreviated form of this command.

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

$Z≔\mathrm{Matrix}\left(\left[\left[x\left(t\right)\,y\left(t\right)\right]\,\left[y\left(t\right)\,-x\left(t\right)\right]\right]\right)$

$Z≔\left[\begin{array}{cc}x\left(t\right)& y\left(t\right)\\ y\left(t\right)& -x\left(t\right)\end{array}\right]$

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

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

$K≔\mathrm{Matrix}\left(\left[\left[c\,0\right]\,\left[0\,c\right]\right]\right)$

$K≔\left[\begin{array}{cc}c& 0\\ 0& c\end{array}\right]$

$\mathrm{Z0}≔\mathrm{Matrix}\left(\left[\left[\mathrm{\_C1}\,\mathrm{\_C2}\right]\,\left[\mathrm{\_C2}\,-\mathrm{\_C1}\right]\right]\right)$

$\mathrm{Z0}≔\left[\begin{array}{cc}\mathrm{\_C1}& \mathrm{\_C2}\\ \mathrm{\_C2}& -\mathrm{\_C1}\end{array}\right]$

$\mathrm{sys}≔\mathrm{map}\left(\mathrm{diff}\,Z\,t\right)=Z·A·Z+Z·K+K^{\mathrm{\%T}}·Z$

$\mathrm{sys}≔\left[\begin{array}{cc}\frac{\ⅆ}{\ⅆt}x\left(t\right)& \frac{\ⅆ}{\ⅆt}y\left(t\right)\\ \frac{\ⅆ}{\ⅆt}y\left(t\right)& -\frac{\ⅆ}{\ⅆt}x\left(t\right)\end{array}\right]=\left[\begin{array}{cc}\left(-x\left(t\right)t+y\left(t\right)\right)x\left(t\right)+\left(x\left(t\right)+y\left(t\right)t\right)y\left(t\right)+2x\left(t\right)c& \left(-x\left(t\right)t+y\left(t\right)\right)y\left(t\right)-\left(x\left(t\right)+y\left(t\right)t\right)x\left(t\right)+2y\left(t\right)c\\ \left(-y\left(t\right)t-x\left(t\right)\right)x\left(t\right)+\left(-x\left(t\right)t+y\left(t\right)\right)y\left(t\right)+2y\left(t\right)c& \left(-y\left(t\right)t-x\left(t\right)\right)y\left(t\right)-\left(-x\left(t\right)t+y\left(t\right)\right)x\left(t\right)-2x\left(t\right)c\end{array}\right]$

$\mathrm{ode}\left[1\right]≔\mathrm{lhs}\left(\mathrm{sys}\right)\left[1,1\right]=\mathrm{rhs}\left(\mathrm{sys}\right)\left[1,1\right]$

${\mathrm{ode}}_1≔\frac{\ⅆ}{\ⅆt}x\left(t\right)=\left(-x\left(t\right)t+y\left(t\right)\right)x\left(t\right)+\left(x\left(t\right)+y\left(t\right)t\right)y\left(t\right)+2x\left(t\right)c$

$\mathrm{ode}\left[2\right]≔\mathrm{lhs}\left(\mathrm{sys}\right)\left[1,2\right]=\mathrm{rhs}\left(\mathrm{sys}\right)\left[1,2\right]$

${\mathrm{ode}}_2≔\frac{\ⅆ}{\ⅆt}y\left(t\right)=\left(-x\left(t\right)t+y\left(t\right)\right)y\left(t\right)-\left(x\left(t\right)+y\left(t\right)t\right)x\left(t\right)+2y\left(t\right)c$

$\mathrm{matrix\_sol}≔\mathrm{matrix\_riccati}\left(A\,K\,\mathrm{Z0}\,t=0\right)\:$

$\mathrm{sol}≔\left\{x\left(t\right)=\mathrm{matrix\_sol}\left[1,1\right]\,y\left(t\right)=\mathrm{matrix\_sol}\left[1,2\right]\right\}$

$\mathrm{sol}≔\left\{x\left(t\right)=\frac{4{\left({\ⅇ}^{tc}\right)}^2{c}^2\left(2{\ⅇ}^{2tc}{\mathrm{\_C1}}^{2}ct+2{\ⅇ}^{2tc}{\mathrm{\_C2}}^{2}ct-{\ⅇ}^{2tc}{\mathrm{\_C1}}^2-{\ⅇ}^{2tc}{\mathrm{\_C2}}^2+4\mathrm{\_C1}{c}^2+{\mathrm{\_C1}}^2+{\mathrm{\_C2}}^2\right)}{4{\left({\ⅇ}^{2tc}\right)}^2{\mathrm{\_C1}}^2{c}^2{t}^2+4{\left({\ⅇ}^{2tc}\right)}^2{\mathrm{\_C2}}^2{c}^2{t}^2+4{\left({\ⅇ}^{2tc}\right)}^2{\mathrm{\_C1}}^2{c}^2-4{\left({\ⅇ}^{2tc}\right)}^2{\mathrm{\_C1}}^{2}ct+4{\left({\ⅇ}^{2tc}\right)}^2{\mathrm{\_C2}}^2{c}^2-4{\left({\ⅇ}^{2tc}\right)}^2{\mathrm{\_C2}}^{2}ct+16{\ⅇ}^{2tc}\mathrm{\_C1}{c}^{3}t-8{\ⅇ}^{2tc}{\mathrm{\_C1}}^2{c}^2+4{\ⅇ}^{2tc}{\mathrm{\_C1}}^{2}ct-8{\ⅇ}^{2tc}{\mathrm{\_C2}}^2{c}^2+4{\ⅇ}^{2tc}{\mathrm{\_C2}}^{2}ct-16{\ⅇ}^{2tc}\mathrm{\_C2}{c}^3+{\left({\ⅇ}^{2tc}\right)}^2{\mathrm{\_C1}}^2+{\left({\ⅇ}^{2tc}\right)}^2{\mathrm{\_C2}}^2-8{\ⅇ}^{2tc}\mathrm{\_C1}{c}^2+4{\mathrm{\_C1}}^2{c}^2+4{\mathrm{\_C2}}^2{c}^2+16\mathrm{\_C2}{c}^3+16c^4-2{\ⅇ}^{2tc}{\mathrm{\_C1}}^2-2{\ⅇ}^{2tc}{\mathrm{\_C2}}^2+8\mathrm{\_C1}{c}^2+{\mathrm{\_C1}}^2+{\mathrm{\_C2}}^2}\,y\left(t\right)=-\frac{8{\left({\ⅇ}^{tc}\right)}^2{c}^3\left({\ⅇ}^{2tc}{\mathrm{\_C1}}^2+{\ⅇ}^{2tc}{\mathrm{\_C2}}^2-{\mathrm{\_C1}}^2-{\mathrm{\_C2}}^2-2\mathrm{\_C2}c\right)}{4{\left({\ⅇ}^{2tc}\right)}^2{\mathrm{\_C1}}^2{c}^2{t}^2+4{\left({\ⅇ}^{2tc}\right)}^2{\mathrm{\_C2}}^2{c}^2{t}^2+4{\left({\ⅇ}^{2tc}\right)}^2{\mathrm{\_C1}}^2{c}^2-4{\left({\ⅇ}^{2tc}\right)}^2{\mathrm{\_C1}}^{2}ct+4{\left({\ⅇ}^{2tc}\right)}^2{\mathrm{\_C2}}^2{c}^2-4{\left({\ⅇ}^{2tc}\right)}^2{\mathrm{\_C2}}^{2}ct+16{\ⅇ}^{2tc}\mathrm{\_C1}{c}^{3}t-8{\ⅇ}^{2tc}{\mathrm{\_C1}}^2{c}^2+4{\ⅇ}^{2tc}{\mathrm{\_C1}}^{2}ct-8{\ⅇ}^{2tc}{\mathrm{\_C2}}^2{c}^2+4{\ⅇ}^{2tc}{\mathrm{\_C2}}^{2}ct-16{\ⅇ}^{2tc}\mathrm{\_C2}{c}^3+{\left({\ⅇ}^{2tc}\right)}^2{\mathrm{\_C1}}^2+{\left({\ⅇ}^{2tc}\right)}^2{\mathrm{\_C2}}^2-8{\ⅇ}^{2tc}\mathrm{\_C1}{c}^2+4{\mathrm{\_C1}}^2{c}^2+4{\mathrm{\_C2}}^2{c}^2+16\mathrm{\_C2}{c}^3+16c^4-2{\ⅇ}^{2tc}{\mathrm{\_C1}}^2-2{\ⅇ}^{2tc}{\mathrm{\_C2}}^2+8\mathrm{\_C1}{c}^2+{\mathrm{\_C1}}^2+{\mathrm{\_C2}}^2}\right\}$

$\mathrm{odetest}\left(\mathrm{sol}\,\left[\mathrm{ode}\left[1\right]\,\mathrm{ode}\left[2\right]\right]\right)$

$\left[0\,0\right]$