\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)
(%i1) kill(all)$

ode : part6,intro: ricatti
[email protected],05/02/2017

Theory Ricatti : ODE (reaction kinetics)

(dy/dx) = p(x)*y^2+q(x)*y+r(x)
Why: Consecutive Reactions and Ricatti
                     k1   k2
reaction kinetics : A → B → C
dCA/dt = - k1*CA^n          (1)
dCB/dt = k1*CA^n-k2*CB^m    (2)
dCC/dt = k2*CC^l            (3)

integer values :  n,m,l
concentration  : CA,CB,CC

(%i2) load('contrib_ode)$
assume(t>0,CA>0,CB>0)$

equation (2) n=2,m=1, by 't=0 → CA=CA0'

(%i5) eq1: CA = CA0/(1+k1*CA0*t);
eq2:'diff(CB,t)=k1*CA^2-k2*CB^1;
eq3:lhs(eq2)-part(eq2,2,2)=rhs(eq2)-part(eq2,2,2);
\[\tag{eq1}\label{eq1}\mathit{CA}=\frac{\mathit{CA0}}{\mathit{CA0}\,\mathit{k1}t+1}\] \[\tag{eq2}\label{eq2}\frac{d}{dt}\mathit{CB}={{\mathit{CA}}^{2}}\,\mathit{k1}-\mathit{CB}\,\mathit{k2}\] \[\tag{eq3}\label{eq3}\mathit{CB}\,\mathit{k2}+\frac{d}{dt}\mathit{CB}={{\mathit{CA}}^{2}}\,\mathit{k1}\]
(%i7) eq4:subst(rhs(eq1),CA,eq3);
eq5: lhs(eq3)=rhs(eq4);
\[\tag{eq4}\label{eq4}\mathit{CB}\,\mathit{k2}+\frac{d}{dt}\mathit{CB}=\frac{{{\mathit{CA0}}^{2}}\,\mathit{k1}}{{{\left( \mathit{CA0}\,\mathit{k1}t+1\right) }^{2}}}\] \[\tag{eq5}\label{eq5}\mathit{CB}\,\mathit{k2}+\frac{d}{dt}\mathit{CB}=\frac{{{\mathit{CA0}}^{2}}\,\mathit{k1}}{{{\left( \mathit{CA0}\,\mathit{k1}t+1\right) }^{2}}}\]
(%i9) eq6:contrib_ode(eq5,CB,t)[1];
method;
\[\tag{eq6}\label{eq6}\mathit{CB}=\left( \mathit{\%{}c}-\frac{\operatorname{expintegral\_ e}\left( 2,-\frac{\mathit{k2}\,\left( \mathit{CA0}\,\mathit{k1}t+1\right) }{\mathit{CA0}\,\mathit{k1}}\right) \mathit{CA0}\,{{\%{}e}^{-\frac{\mathit{k2}}{\mathit{CA0}\,\mathit{k1}}}}}{\mathit{CA0}\,\mathit{k1}t+1}\right) {{\%{}e}^{-\mathit{k2}t}}\] \[\tag{\%{}o9}\label{o9} \mathit{linear}\]

function :build in maxima = 'expintegral_e(r,z)'
info :

https://en.wikipedia.org/wiki/Exponential_integral
http://maxima-online.org/help/index/expintegral_e
http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/
http://encyclopedia2.thefreedictionary.com/Exponential+integral

(%i10) eq7:expintegral_e(2,-(k2*(CA0*k1*t+1))/(CA0*k1));
\[\tag{eq7}\label{eq7}\operatorname{expintegral\_ e}\left( 2,-\frac{\mathit{k2}\,\left( \mathit{CA0}\,\mathit{k1}t+1\right) }{\mathit{CA0}\,\mathit{k1}}\right) \]

find %c,by t=0 and CB=0 in eq6

(%i12) eq8:ev(eq6,t=0,CB=0);
eq9:solve(eq8,%c)[1];
\[\tag{eq8}\label{eq8}0=\mathit{\%{}c}-\operatorname{expintegral\_ e}\left( 2,-\frac{\mathit{k2}}{\mathit{CA0}\,\mathit{k1}}\right) \mathit{CA0}\,{{\%{}e}^{-\frac{\mathit{k2}}{\mathit{CA0}\,\mathit{k1}}}}\] \[\tag{eq9}\label{eq9}\mathit{\%{}c}=\operatorname{expintegral\_ e}\left( 2,-\frac{\mathit{k2}}{\mathit{CA0}\,\mathit{k1}}\right) \mathit{CA0}\,{{\%{}e}^{-\frac{\mathit{k2}}{\mathit{CA0}\,\mathit{k1}}}}\]

solution : n=1,m=2 for eq5

(%i13) eq10:ev(eq6,eq9);
\[\tag{eq10}\label{eq10}\mathit{CB}=\left( \operatorname{expintegral\_ e}\left( 2,-\frac{\mathit{k2}}{\mathit{CA0}\,\mathit{k1}}\right) \mathit{CA0}\,{{\%{}e}^{-\frac{\mathit{k2}}{\mathit{CA0}\,\mathit{k1}}}}-\frac{\operatorname{expintegral\_ e}\left( 2,-\frac{\mathit{k2}\,\left( \mathit{CA0}\,\mathit{k1}t+1\right) }{\mathit{CA0}\,\mathit{k1}}\right) \mathit{CA0}\,{{\%{}e}^{-\frac{\mathit{k2}}{\mathit{CA0}\,\mathit{k1}}}}}{\mathit{CA0}\,\mathit{k1}t+1}\right) {{\%{}e}^{-\mathit{k2}t}}\]
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