November 3rd, 2015, 07:56 PM  #1 
Newbie Joined: Nov 2015 From: Chicago Posts: 2 Thanks: 0  Power series on ODE solution
Dear all, I am a bit rusty on power series expansion, and I am stucked with the following problem. I have the following ODE in terms of two continuous differentiable generic functions $\displaystyle u(t)$ and $\displaystyle v(t)$: $\displaystyle \dot{u} = g_L u + g_D (v(t)u(t)) + g(t) (Eu)$ where $\displaystyle g_L,g_D,E$ real and >0 and $\displaystyle g(t)$ is continuous and nonnegative. The solution outline in my book suggests that under the assumption that $\displaystyle g_D \gg g_L$ it is possible to expand in power series of $\displaystyle 1/g_D$ and obtain up to the leading order: $\displaystyle u = (1\lambda)E + \lambda v* + O(1/g_D)$ where: $\displaystyle \lambda = (g_D+g_L)/g_T$ $\displaystyle g_T = g_L+g_D+g(t)$ $\displaystyle v* = g_D/(g_L+g_D)$ I cannot figure out the practical steps to obtain such result. Any help would be very welcome! Thanks in advance! M 
November 4th, 2015, 04:09 AM  #2 
Newbie Joined: Nov 2015 From: Chicago Posts: 2 Thanks: 0  The way is by integration by parts
There is a straight forward solution to my question, after a bit of playing around. The solution of the ODE is in fact: $\displaystyle u(t) = e^{(g_L+g_D)t}\int_0^t dz\, g_D v(z) e^{(g_L+g_D)z}$. Then one notes that the integral on the rhs of the above equation may be solved by integration by parts whereby $\displaystyle \int_0^t d\left(v(z) e^{(g_L+g_D)z}\right) = \int_0^t dz\, v'(z)e^{(g_L+g_D)z} + \frac{g_D}{g_D+g_L}\int_0^t v(z)d\left(e^{(g_D+g_L)z}\right)$ $\displaystyle \Rightarrow \int_0^t v(z)d\left(e^{(g_D+g_L)z}\right) = \frac{g_D}{g_D+g_L} \left( v(t)e^{(g_D+g_L)t} + \int_0^t dz\, v'(t)e^{(g_D+g_L)z}\right)$. By replacing the latter in the solution, and ordering terms by powers of $\displaystyle 1/g_D$ it is straight forward to see that, up to the leading order, it is $\displaystyle u(t) = e^{(g_L+g_D)t} \frac{g_D}{g_D+g_L} \left( v(t)e^{(g_D+g_L)t} + \int_0^t dz\, v'(t)e^{(g_D+g_L)z}\right)$ $\displaystyle \Rightarrow u(t) = \frac{g_D}{g_D+g_L} v(t) + O\left(\frac{1}{g_D}\right)$. Last edited by skipjack; November 4th, 2015 at 06:33 AM. 

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ode, power, series, solution 
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