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

ode : part1,intro: ricatti
[email protected],03/02/2017

Theory Ricatti : ODE (reaction kinetics)

(dy/dx) = p(x)*y^2+q(x)*y+r(x)
Why: Consective 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)$

example: (1) n=1,2
y = CA,x = t
n=1 ⇒ p(x) = 0,r(x)=0,q(x)<>0
n=2 ⇒ q(x) = 0,r(x)=0,p(x)<>0

n=1,equation (1),initial value CA = CA0

(%i7) eq1:'diff(CA,t) = - k1*CA^1;
assume(CA>0)$
eq2:contrib_ode(eq1,CA,t)[1]$
method;
eq3:subst(CA0,%c,eq2);
\[\tag{eq1}\label{eq1}\frac{d}{dt}\mathit{CA}=-\mathit{CA}\,\mathit{k1}\] \[\tag{\%{}o6}\label{o6} \mathit{linear}\] \[\tag{eq3}\label{eq3}\mathit{CA}=\mathit{CA0}\,{{\%{}e}^{-\mathit{k1}t}}\]

n=2,equation (1),initial value CA = CA0

first method :

(%i15) eq4:'diff(CA,t) = - k1*CA^2;
assume(CA>0)$
eq5:contrib_ode(eq4,CA,t)[1]$
method;
eq6:subst(CA0,%c,eq5);
eq7:factor(solve(eq6,CA)[1]);
eq8:factor(ratsubst(rhs(eq7),CA,eq4));
eq81:part(eq8,1);
\[\tag{eq4}\label{eq4}\frac{d}{dt}\mathit{CA}=-{{\mathit{CA}}^{2}}\,\mathit{k1}\] \[\tag{\%{}o11}\label{o11} \mathit{separable}\] \[\tag{eq6}\label{eq6}\frac{1}{\mathit{CA}\,\mathit{k1}}=t+\mathit{CA0}\] \[\tag{eq7}\label{eq7}\mathit{CA}=\frac{1}{\mathit{k1}\,\left( t+\mathit{CA0}\right) }\] \[\tag{eq8}\label{eq8}\frac{d}{dt}\frac{1}{\mathit{k1}\,\left( t+\mathit{CA0}\right) }=-\frac{1}{\mathit{k1}\,{{\left( t+\mathit{CA0}\right) }^{2}}}\] \[\tag{eq81}\label{eq81}\frac{d}{dt}\frac{1}{\mathit{k1}\,\left( t+\mathit{CA0}\right) }\]

check solution for: n=2

(%i16) eq9:ev(part(eq8,1),nouns)=ev(part(eq8,2),nouns);
\[\tag{eq9}\label{eq9}-\frac{1}{\mathit{k1}\,{{\left( t+\mathit{CA0}\right) }^{2}}}=-\frac{1}{\mathit{k1}\,{{\left( t+\mathit{CA0}\right) }^{2}}}\]

appendix: command: ev,nouns,part

(%i17) ev('diff(1/(k1*(t+CA0)),t,1), nouns);
\[\tag{\%{}o17}\label{o17} -\frac{1}{\mathit{k1}\,{{\left( t+\mathit{CA0}\right) }^{2}}}\]
(%i18) eqa1:'diff(CA,t,1) = - k1*CA^2;
\[\tag{eqa1}\label{eqa1}\frac{d}{dt}\mathit{CA}=-{{\mathit{CA}}^{2}}\,\mathit{k1}\]
(%i19) part(eqa1,1,2);
\[\tag{\%{}o19}\label{o19} t\]

second method : use ode2

(%i22) ode2('diff(CA,t,1)=-CA^2*k1, CA, t);
subst(CA0,%c,%);
factor(solve(%,CA))[1];
\[\tag{\%{}o20}\label{o20} \frac{1}{\mathit{CA}\,\mathit{k1}}=t+\mathit{\%{}c}\] \[\tag{\%{}o21}\label{o21} \frac{1}{\mathit{CA}\,\mathit{k1}}=t+\mathit{CA0}\] \[\tag{\%{}o22}\label{o22} \mathit{CA}=\frac{1}{\mathit{k1}\,\left( t+\mathit{CA0}\right) }\]

third method :manual:separable

(%i29) eq12:integrate(CA^(-2), CA);
eq13:-integrate(k1,t);
eq14:eq12=eq13;
eq15:ratsubst(-1/CA,-1/CA,lhs(eq14));
eq16:ratsubst(-1/CA0,-1/CA,lhs(eq14));
eq17:eq15-eq16=eq13;
eq18:solve(eq17,CA)[1];
\[\tag{eq12}\label{eq12}-\frac{1}{\mathit{CA}}\] \[\tag{eq13}\label{eq13}-\mathit{k1}t\] \[\tag{eq14}\label{eq14}-\frac{1}{\mathit{CA}}=-\mathit{k1}t\] \[\tag{eq15}\label{eq15}-\frac{1}{\mathit{CA}}\] \[\tag{eq16}\label{eq16}-\frac{1}{\mathit{CA0}}\] \[\tag{eq17}\label{eq17}\frac{1}{\mathit{CA0}}-\frac{1}{\mathit{CA}}=-\mathit{k1}t\] \[\tag{eq18}\label{eq18}\mathit{CA}=\frac{\mathit{CA0}}{\mathit{CA0}\,\mathit{k1}t+1}\]
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