# Technique on solving non-linear ODE

1. May 21, 2007

### yogi10usa

I have a set of non-linear ODE.
There are eight variables that depends on time (t)
4 of those are first order ODE
1 of those are first order non-linear ODE
3 of those are non-linear equations

The variables are
(t)

(p)
(ps)
(gs)
(gd)
(as)
(hs)
(cs)

k1,k2,k3,k4,k5 are constants

The following initial condition are given at t=0
(p)
(gs)
(gd)
(as)

The diff equations are

note: f(t) is a set a data defined from t =0 from t=1000

#1: d(p)/d(t) = (ps)-(p) + f(t)

#2: d(gs)/d(t) = (gd)-(gs)-(ps)-(p)-k3

#3: d(gd)/d(t) = (gd) - (gs)-k1

#4: d(gd)/d(t) =(gd)-(gs) -k2

#5: d(gd)/d(t) = (gd)-(gs)-k4

The non-linear equations are

#6: (ps) = ((hs)^2)/(cs)

#7: (cs) = ((as)-(hs))/2

#8: (hs) = (gs) - {(gs)^2 - k5*(as)*[2*(gs)-(as)]}^(1/2)

I figure there are two approaches so far

1st approach:)
I figure I can figure differentiate #6, #7, #8 and figure out their initial condition and use ODE45 or ODE23 to solve this system

However, #8 will be very messy......

2nd approach:)
I can solve for (ps) using #6, #7 and #8 and thus I have a system of
5 non-linear ODE. (#1 - #5) It is still very messy......

Any ideas in this kind of situation???

Also, what do I need to do for f(t) as it is given as a set of points and not a well-defined function.

Thanks so much for your help