Using the maple I am trying to get quardic in q from this big linear equation. Then use Descarte’s rule of signs to determine the number of positive roots.
\begin{equation}
\frac{\gamma*q*P_Q}{k_p*(1-q)*P_C} = \frac{I*\alpha}{k_f+k_d+\frac{k_n*\lambda_b*\gamma *q*P_Q}{\lambda_b*\gamma*q*P_Q+k_p*\lambda_r*(1-q)^2}+k_p*(1-q)}
\end{equation}
Values of parameters are given below:
$I=1200$
$k_f = 6.7*10.^7$
$k_d = 6.03*10.^8$
$k_n = 2.92*10.^9$
$k_p = 4.94*10.^9$
$\alpha = 1.14437*10.^(-3)$
$\lambda_b = 0.87e-2$
$\lambda_r = 835$
$\gamma = 2.74$
$P_C = 3*10.^(11)$
$P_Q = 2.87*10.^(10)$
=>
I tried the code in maple to get quartic in q but DOES NOT WORKS.
Code:
II := 1200:
k_f := 6.7*10.^7:
k_d := 6.03*10.^8:
k_n := 2.92*10.^9:
k_p := 4.94*10.^9:
alpha := 1.14437*10.^(-3):
lambda_b := 0.87e-2:
lambda_r := 835:
ggamma := 2.74:
P_C := 3*10.^11:
P_Q := 2.87*10.^10:
eq := ggamma*q*P_Q/(k_p*(1-q)*P_C) = II*alpha/(k_f+k_d+k_n*lambda_b*ggamma*q*P_Q/(lambda_b*ggamma*q*P_Q+k_p*lambda_r*(1-q)^2)+k_p*(1-q)):
simply(eq, q);
My lecturer want me to manipulate the equation and get a quartic in q before substituting the values of parameters into the equation. After that,use Descarte’s rule of signs to determine the number of positive roots. Then write Q=1-q to get second quartic in Q and repeat rule of signs to determine number of steady states of q less than 1. And do the substition of parameters if necessary.
Now, its kind of hard for me what he wants because to get quartic in q first from the equation is hard to do by hand , so i have to use in maple which is not working then use Descarte’s rule of signs.