By using the second order ODE and its solution how i can get thus changing Rule?  i am looking for equation (8) by assumption of the paper he did, there is a trick but i don't know

idea.mw

This expression is easy i think but i can't get any results with same shape  how i can simplify to the same equation in the paper

i am looking for equation 9 and every thing is clear how i can get

mp1.mw

i have solution of ODE but again i want take derivative from solution function F then i want take reciprocal of derivative
if F'=G then i want 1/F'=1/G like that i want all solution by list and if possible don't give the parameter a sequence  it will be better

thanks for any help

K := diff(G(xi), xi $ 2) = -lambda*diff(G(xi), xi) - mu;
                 2                                    
                d                    / d        \     
          K := ----- G(xi) = -lambda |---- G(xi)| - mu
                   2                 \ dxi      /     
                dxi                                   

V:= [seq](-1..1, 1/2);
                          [    -1     1   ]
                     V := [-1, --, 0, -, 1]
                          [    2      2   ]

interface(rtablesize= nops(V)^3):
DataFrame(
    <seq(seq(<a | b | rhs(dsolve(eval(K, [lambda,mu]=~ [a,b])))>, a= V), b= V)>,
    columns= [lambda, mu, F]
);

PDF ready for download.

Error occurred during PDF generation. Please refresh the page and try again

Hi
my odetest must give me zero everything is true but still not simplify the function with power include 1/n  not do cancelation even

test_sol_for_PDE.mw

Hi
most of my equation when i wanna solve it is give me something like thus equation and it take a lot time to calculate and i don't get any answer after 1 hour past and still searching for finding parameter how i can do in fast way or maybe if exist  can i use mathematica or matlab is better for calculating this or not ?

F-P.mw