riccati_system can find a closed-form solution for certain systems of riccati ODEs. A system of riccati ODEs of n variables is:

with initial conditions

where $x_1..x_n$ are the unknown functions and $P_1..P_n$ are polynomials in $x_1..x_n$ of second degree with function coefficients.

To solve this system, riccati_system tries to convert it into a matrix equation,

with

where $Z,A,K$, and $\mathrm{Z0}$ are n by n matrices. The user can request this matrix form instead of a full solution by using the 'matrix_only' option. In this case a sequence $Z,A,K,\mathrm{Z0},t_0$ is returned.

This function is part of the DEtools package, and so it can be used in the form riccati_system(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[riccati_system](..).

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

$\mathrm{e1}≔t\left({y\left(t\right)}^2-{x\left(t\right)}^2\right)+2x\left(t\right)y\left(t\right)+2cx\left(t\right)=\mathrm{diff}\left(x\left(t\right)\,t\right)$

$\mathrm{e1}≔t\left({y\left(t\right)}^2-{x\left(t\right)}^2\right)+2x\left(t\right)y\left(t\right)+2cx\left(t\right)=\frac{\ⅆ}{\ⅆt}x\left(t\right)$

$\mathrm{e2}≔{y\left(t\right)}^2-{x\left(t\right)}^2-t\cdot 2x\left(t\right)y\left(t\right)+2cy\left(t\right)=\mathrm{diff}\left(y\left(t\right)\,t\right)$

$\mathrm{e2}≔{y\left(t\right)}^2-{x\left(t\right)}^2-2tx\left(t\right)y\left(t\right)+2cy\left(t\right)=\frac{\ⅆ}{\ⅆt}y\left(t\right)$

$\mathrm{sol}≔\mathrm{riccati\_system}\left(\left\{\mathrm{e1}\,\mathrm{e2}\,x\left(0\right)=\mathrm{\_C1}\,y\left(0\right)=\mathrm{\_C2}\right\}\,\left[x\left(t\right)\,y\left(t\right)\right]\right)$

$\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\}$

$\mathrm{odetest}\left(\mathrm{sol}\,\left[\mathrm{e1}\,\mathrm{e2}\right]\right)$

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

$Z,A,K,\mathrm{Z0},\mathrm{t0}≔\mathrm{riccati\_system}\left(\left\{\mathrm{e1}\,\mathrm{e2}\,x\left(0\right)=\mathrm{\_C1}\,y\left(0\right)=\mathrm{\_C2}\right\}\,\left[x\left(t\right)\,y\left(t\right)\right]\,\mathrm{matrix\_only}\right)$

$Z,A,K,\mathrm{Z0},\mathrm{t0}≔\left[\begin{array}{cc}x\left(t\right)& y\left(t\right)\\ y\left(t\right)& -x\left(t\right)\end{array}\right],\left[\begin{array}{cc}-t& 1\\ 1& t\end{array}\right],\left[\begin{array}{cc}c& 0\\ 0& c\end{array}\right],\left[\begin{array}{cc}\mathrm{\_C1}& \mathrm{\_C2}\\ \mathrm{\_C2}& -\mathrm{\_C1}\end{array}\right],0$

$\mathrm{RSM\_sol}≔\mathrm{matrix\_riccati}\left(A\,K\,\mathrm{Z0}\,t=\mathrm{t0}\right)\:$

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

$\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\}$

$\mathrm{odetest}\left(\mathrm{sol}\,\left[\mathrm{e1}\,\mathrm{e2}\right]\right)$

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