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](..).

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

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

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

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

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

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

$\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\}$

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

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

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

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

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

$\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\}$

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

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