Benutzer:Dirk Huenniger/med

${\begin{array}{ccc}q_{1}(t=0)&=&1\\q_{2}(t=0)&=&0\\q_{3}(t=0)&=&0\end{array}}$

${\begin{array}{ccc}{\overset {\text{ }}{\frac {\mathrm {d} q_{1}}{\mathrm {d} t}}}&=&-\left(k_{13}+k_{12}\right)\cdot q_{1}+k_{31}q_{3}\\{\overset {\text{ }}{\frac {\mathrm {d} q_{2}}{\mathrm {d} t}}}&=&k_{12}\cdot q_{1}\\{\overset {\text{ }}{\frac {\mathrm {d} q_{3}}{\mathrm {d} t}}}&=&k_{13}\cdot q_{1}-k_{31}q_{3}\end{array}}$

${\begin{array}{ccc}q_{1}&=&A_{1}e^{l_{1}t}+A_{2}e^{l_{2}t}\\q_{2}&=&1-A_{3}e^{l_{1}t}-A_{4}e^{l_{2}t}\\q_{3}&=&-A_{5}e^{l_{1}t}+A_{5}e^{l_{2}t}\end{array}}$

${\begin{array}{ccc}{\overset {\text{ }}{\frac {\mathrm {d} q_{1}}{\mathrm {d} t}}}&=&l_{1}A_{1}e^{l_{1}t}+l_{2}A_{2}e^{l_{2}t}\\{\overset {\text{ }}{\frac {\mathrm {d} q_{2}}{\mathrm {d} t}}}&=&-l_{1}A_{3}e^{l_{1}t}-l_{2}A_{4}e^{l_{2}t}\\{\overset {\text{ }}{\frac {\mathrm {d} q_{3}}{\mathrm {d} t}}}&=&-l_{1}A_{5}e^{l_{1}t}+l_{2}A_{5}e^{l_{2}t}\end{array}}$

${\begin{array}{ccc}-\left(k_{13}+k_{12}\right)\cdot q_{1}+k_{31}q_{3}&=&-\left(k_{13}+k_{12}\right)\left(A_{1}e^{l_{1}t}+A_{2}e^{l_{2}t}\right)+k_{31}\left(-A_{5}e^{l_{1}t}+A_{2}e^{l_{2}t}\right)\\k_{12}\cdot q_{1}&=&k_{12}\left(A_{1}e^{l_{1}t}+A_{2}e^{l_{2}t}\right)\\k_{13}\cdot q_{1}-k_{31}q_{3}&=&k_{13}\cdot \left(A_{1}e^{l_{1}t}+A_{2}e^{l_{2}t}\right)-k_{31}\left(-A_{5}e^{l_{1}t}+A_{5}e^{l_{2}t}\right)\end{array}}$

${\begin{array}{ccc}l_{1}A_{1}&=&-\left(k_{13}+k_{12}\right)A_{1}-k_{31}A_{5}\\l_{2}A_{2}&=&-\left(k_{13}+k_{12}\right)A_{2}+k_{31}A_{5}\\-l_{1}A_{3}&=&k_{12}A_{1}\\-l_{2}A_{4}&=&k_{12}A_{2}\\-l_{1}A_{5}&=&k_{13}A_{1}+k_{31}A_{5}\\-l_{2}A_{5}&=&k_{13}A_{2}-k_{31}A_{5}\\1&=&A_{1}+A_{2}\\0&=&1-A_{3}-A_{4}\\0&=&-A_{5}+A_{5}\end{array}}$

${\begin{array}{ccc}l_{1}&=&-{\frac {k_{12}A_{1}}{A_{3}}}\\l_{2}&=&-{\frac {k_{12}A_{2}}{A_{4}}}\\1&=&A_{1}+A_{2}\\1&=&A_{3}+A_{4}\\\end{array}}$

${\begin{array}{ccc}k_{12}A_{1}^{2}&=&-\left(k_{13}+k_{12}\right)A_{1}A_{3}-k_{31}A_{5}A_{3}\\k_{12}A_{2}^{2}&=&-\left(k_{13}+k_{12}\right)A_{2}A_{4}+k_{31}A_{5}A_{4}\\\end{array}}$

${\begin{array}{ccc}k_{12}A_{1}^{2}&=&-\left(k_{13}+k_{12}\right)A_{1}A_{3}-k_{31}A_{2}A_{3}\\A_{2}&=&{\frac {-\left(k_{13}+k_{12}\right)A_{4}+k_{31}A_{4}}{k_{12}}}\\A_{1}&=&1-A_{2}\\A_{4}&=&\left(1-A_{3}\right)\\\end{array}}$

${\begin{array}{ccc}k_{12}A_{1}^{2}&=&-\left(k_{13}+k_{12}\right)A_{1}A_{3}-k_{31}A_{2}A_{3}\\A_{2}&=&{\overset {\text{ }}{\frac {-\left(k_{13}+k_{12}\right)\left(1-A_{3}\right)+k_{31}\left(1-A_{3}\right)}{k_{12}}}}\\A_{1}&=&{\overset {\text{ }}{1+{\frac {\left(k_{13}+k_{12}\right)\left(1-A_{3}\right)-k_{31}\left(1-A_{3}\right)}{k_{12}}}}}\\\end{array}}$

${\begin{array}{ccc}c_{10}&:=&{\frac {k_{13}+2k_{12}-k_{31}}{k_{12}}}\\c_{11}&:=&{\overset {\text{ }}{\frac {k_{31}-k_{13}-k_{12}}{k_{12}}}}\\c_{20}&:=&{\overset {\text{ }}{\frac {-k_{13}-k_{12}+k_{31}}{k_{12}}}}\\c_{21}&:=&{\overset {\text{ }}{\frac {k_{13}+k_{12}-k_{31}}{k_{12}}}}\\A_{1}&=&c_{10}+c_{11}A_{3}\\A_{2}&=&c_{20}+c_{21}A_{3}\end{array}}$

$k_{12}\left(c_{10}^{2}+2c_{10}c_{11}A_{3}+c_{11}^{2}A_{3}^{2}\right)=-\left(c_{10}k_{13}+c_{10}k_{12}\right)A_{3}-\left(c_{11}k_{13}+c_{11}k_{12}\right)A_{3}^{2}-k_{31}c_{20}A_{3}-k_{31}c_{21}A_{3}^{2}$

$k_{12}c_{10}^{2}+\left(2k_{12}c_{10}c_{11}+c_{10}k_{13}+c_{10}k_{12}+k_{31}c_{20}\right)A_{3}+\left(c_{11}^{2}k_{12}+c_{11}k_{13}+c_{11}k_{12}+k_{31}c_{21}\right)A_{3}^{2}=0$

${\begin{array}{ccc}k_{12}&=&0.05\\k_{13}&=&0.04\\k_{31}&=&0.06\end{array}}$