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Math Help - Asymptotic expansion on 3 nonlinear ordinary differential equations

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    Asymptotic expansion on 3 nonlinear ordinary differential equations

    The 3 nonlinear differential equations are as follows
    \begin{equation}
    \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber
    \end{equation}
    \begin{equation}
    \frac{ds}{dt}= \lambda_b P_C \ \epsilon \ c (1-s)- \lambda_r (1-q) \ s, \nonumber
    \end{equation}
    \begin{equation}
    \frac{dq}{dt}= K_P (1-q) \frac{P_C}{P_Q} \ \ c - \gamma \ q, \nonumber
    \end{equation}
    I want to use asymptotic expansion on $c, s$ and $q$.
    And values of parameters are:
    $K_F = 6.7 \times 10^{-2},$
    $K_N = 6.03 \times 10^{-1}$
    $K_P = 2.92 \times 10^{-2}$,
    $K_D = 4.94 \times 10^{-2}$,
    $\lambda_b= 0.0087$,
    $I=1200$
    $P_C = 3 \times 10^{11}$
    $P_Q = 2.304 \times 10^{9}$
    $\gamma=2.74 $
    $\lambda_{b}=0.0087 $
    $\lambda_{r}= 835$
    $\alpha=1.14437 \times 10^{-3}$
    For initial conditions:
    \begin{equation}
    c_0(0)= c(0) = 0.25 \nonumber
    \end{equation}
    \begin{equation}
    s_0(0)= cs(0) = 0.02 \nonumber \nonumber
    \end{equation}
    \begin{equation}
    q_0(0)=q(0) = 0.98 \nonumber \nonumber
    \end{equation}
    and
    \begin{equation}
    c_i(0)= 0, \ i>0\nonumber
    \end{equation}
    \begin{equation}
    s_i(0)= 0, \ i>0 \nonumber \nonumber
    \end{equation}
    \begin{equation}
    q_i(0)=0, i>0. \nonumber \nonumber
    \end{equation}
    => i started with the expansions :
    \begin{equation}
    c= c_0+ \epsilon c_1 + \epsilon^2 c_2+......... \nonumber
    \end{equation}
    \begin{equation}
    s= s_0+ \epsilon s_1 + \epsilon^2 s_2+......... \nonumber
    \end{equation}
    \begin{equation}
    q= q_0+ \epsilon q_1 + \epsilon^2 q_2+......... \nonumber
    \end{equation}
    we are only interseted in up to fisrt power of $\epsilon$.
    so, we should get total 6 approximate differential equations to get answer for
    $\frac{dc_0}{dt}, \frac{ds_0}{dt}, \frac{dq_0}{dt}, \frac{dc_1}{dt}, \frac{ds_1}{dt}$ and $\frac{dq_1}{dt}$
    but i think $\frac{dc_1}{dt}$ will disappear while expanding and equating the up to first power of $\epsilon$, do i need to go further up to $\epsilon{^2}$ because $\frac{dc_1}{dt}$ is very important to find and we need 6 approximate differetial equations in total. what can i do? please some one help me.
    Last edited by grandy; July 25th 2014 at 01:52 PM.
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