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(%i1) kill(all)$

ode : part2,intro: ricatti
[email protected],03/02/2017

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=1,m=1, by 't=0 → CA=CA0'

(%i4) eq1: CA = CA0*%e^(-k1*t);
eq2:'diff(CB,t)=k1*CA^1-k2*CB^1;
\[\tag{eq1}\label{eq1}\mathit{CA}=\mathit{CA0}\,{{\%{}e}^{-\mathit{k1}t}}\] \[\tag{eq2}\label{eq2}\frac{d}{dt}\mathit{CB}=\mathit{CA}\,\mathit{k1}-\mathit{CB}\,\mathit{k2}\]

first order ODE : dCB/dt+g(t)*CB = f(t)

method1:

(%i7) eq3:subst(rhs(eq1),CA,eq2);
eq4:part(eq3,2,2);
eq5: lhs(eq3)-eq4=rhs(eq3)-eq4;
\[\tag{eq3}\label{eq3}\frac{d}{dt}\mathit{CB}=\mathit{CA0}\,\mathit{k1}\,{{\%{}e}^{-\mathit{k1}t}}-\mathit{CB}\,\mathit{k2}\] \[\tag{eq4}\label{eq4}-\mathit{CB}\,\mathit{k2}\] \[\tag{eq5}\label{eq5}\mathit{CB}\,\mathit{k2}+\frac{d}{dt}\mathit{CB}=\mathit{CA0}\,\mathit{k1}\,{{\%{}e}^{-\mathit{k1}t}}\]
(%i8) eq6:contrib_ode(eq5,CB,t)[1];
\[\tag{eq6}\label{eq6}\mathit{CB}={{\%{}e}^{-\mathit{k2}t}}\,\left( \frac{\mathit{CA0}\,\mathit{k1}\,{{\%{}e}^{\mathit{k2}t-\mathit{k1}t}}}{\mathit{k2}-\mathit{k1}}+\mathit{\%{}c}\right) \]

assume CB=0 by 't=0 find %c'

(%i10) eq7:ev(eq6,t=0,CB=0);
eq8:solve(eq7,%c)[1];
\[\tag{eq7}\label{eq7}0=\frac{\mathit{CA0}\,\mathit{k1}}{\mathit{k2}-\mathit{k1}}+\mathit{\%{}c}\] \[\tag{eq8}\label{eq8}\mathit{\%{}c}=-\frac{\mathit{CA0}\,\mathit{k1}}{\mathit{k2}-\mathit{k1}}\]
(%i11) eq9:subst(rhs(eq8),%c,eq6);
\[\tag{eq9}\label{eq9}\mathit{CB}={{\%{}e}^{-\mathit{k2}t}}\,\left( \frac{\mathit{CA0}\,\mathit{k1}\,{{\%{}e}^{\mathit{k2}t-\mathit{k1}t}}}{\mathit{k2}-\mathit{k1}}-\frac{\mathit{CA0}\,\mathit{k1}}{\mathit{k2}-\mathit{k1}}\right) \]
(%i12) eq10:radcan(eq9);
\[\tag{eq10}\label{eq10}\mathit{CB}=\frac{\left( \mathit{CA0}\,\mathit{k1}\,{{\%{}e}^{\mathit{k2}t}}-\mathit{CA0}\,\mathit{k1}\,{{\%{}e}^{\mathit{k1}t}}\right) \,{{\%{}e}^{-\left( \mathit{k2}+\mathit{k1}\right) t}}}{\mathit{k2}-\mathit{k1}}\]
(%i13) eq11:factor(part(eq10,2,1));
\[\tag{eq11}\label{eq11}\mathit{CA0}\,\mathit{k1}\,\left( {{\%{}e}^{\mathit{k2}t}}-{{\%{}e}^{\mathit{k1}t}}\right) \,{{\%{}e}^{-\mathit{k2}t-\mathit{k1}t}}\]
(%i14) eq12:lhs(eq10)=eq11/part(eq10,2,2);
\[\tag{eq12}\label{eq12}\mathit{CB}=\frac{\mathit{CA0}\,\mathit{k1}\,\left( {{\%{}e}^{\mathit{k2}t}}-{{\%{}e}^{\mathit{k1}t}}\right) \,{{\%{}e}^{-\mathit{k2}t-\mathit{k1}t}}}{\mathit{k2}-\mathit{k1}}\]

check : solution n=1,m=1

(%i15) eq13:ratsubst(rhs(eq12),CB,eq3);
\[\tag{eq13}\label{eq13}\frac{d}{dt}\frac{\mathit{CA0}\,\mathit{k1}\,\left( {{\%{}e}^{\mathit{k2}t}}-{{\%{}e}^{\mathit{k1}t}}\right) \,{{\%{}e}^{-\mathit{k2}t-\mathit{k1}t}}}{\mathit{k2}-\mathit{k1}}=\frac{{{\%{}e}^{-\mathit{k1}t}}\,\left( \mathit{CA0}\,\mathit{k1}\,\mathit{k2}\,{{\%{}e}^{\mathit{k1}t}}-\mathit{CA0}\,{{\mathit{k1}}^{2}}\,{{\%{}e}^{\mathit{k2}t}}\right) }{\mathit{k2}\,{{\%{}e}^{\mathit{k2}t}}-\mathit{k1}\,{{\%{}e}^{\mathit{k2}t}}}\]
(%i16) eq14:ev(part(eq13,1),nouns)=ev(part(eq13,2),nouns);
\[\tag{eq14}\label{eq14}\frac{\mathit{CA0}\,\mathit{k1}\,\left( \mathit{k2}\,{{\%{}e}^{\mathit{k2}t}}-\mathit{k1}\,{{\%{}e}^{\mathit{k1}t}}\right) \,{{\%{}e}^{-\mathit{k2}t-\mathit{k1}t}}}{\mathit{k2}-\mathit{k1}}+\frac{\mathit{CA0}\,\mathit{k1}\,\left( -\mathit{k2}-\mathit{k1}\right) \,\left( {{\%{}e}^{\mathit{k2}t}}-{{\%{}e}^{\mathit{k1}t}}\right) \,{{\%{}e}^{-\mathit{k2}t-\mathit{k1}t}}}{\mathit{k2}-\mathit{k1}}=\frac{{{\%{}e}^{-\mathit{k1}t}}\,\left( \mathit{CA0}\,\mathit{k1}\,\mathit{k2}\,{{\%{}e}^{\mathit{k1}t}}-\mathit{CA0}\,{{\mathit{k1}}^{2}}\,{{\%{}e}^{\mathit{k2}t}}\right) }{\mathit{k2}\,{{\%{}e}^{\mathit{k2}t}}-\mathit{k1}\,{{\%{}e}^{\mathit{k2}t}}}\]
(%i17) eq5:radcan(eq14);
\[\tag{eq5}\label{eq5}-\frac{\left( \mathit{CA0}\,{{\mathit{k1}}^{2}}\,{{\%{}e}^{\mathit{k2}t}}-\mathit{CA0}\,\mathit{k1}\,\mathit{k2}\,{{\%{}e}^{\mathit{k1}t}}\right) \,{{\%{}e}^{-\left( \mathit{k2}+\mathit{k1}\right) t}}}{\mathit{k2}-\mathit{k1}}=-\frac{\left( \mathit{CA0}\,{{\mathit{k1}}^{2}}\,{{\%{}e}^{\mathit{k2}t}}-\mathit{CA0}\,\mathit{k1}\,\mathit{k2}\,{{\%{}e}^{\mathit{k1}t}}\right) \,{{\%{}e}^{-\left( \mathit{k2}+\mathit{k1}\right) t}}}{\mathit{k2}-\mathit{k1}}\]
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