@janhardo  i put condition on it but didn't work it and i don't know how he reach this goal.

@janhardo  but again i didn't get result

ode1 := diff(G(xi), xi)^2 = A^2 + 2*A*B*G(xi) + 2*A*C*G(xi)^2 + 2*B*C*G(xi)^3 + C^2*G(xi)^4;
                              /         
ode1 := Typesetting:-mcomplete|G[ξ], 
                              \         

                    /[ d        ]\\      2
  Typesetting:-_Hold|[---- G(xi)]||^2 = A 
                    \[ dxi      ]//       

   + 2 A B Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)]))

   + 2 A C 

                                                        2
  Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)])) 

   + 2 B C 

                                                        3
  Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)])) 

      2                                                       4
   + C  Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)])) 


W := B = -2*A*C;
                        W := B = -2 A C

ode2 := subs(W, ode1);
                              /         
ode2 := Typesetting:-mcomplete|G[ξ], 
                              \         

                    /[ d        ]\\      2
  Typesetting:-_Hold|[---- G(xi)]||^2 = A 
                    \[ dxi      ]//       

        2                                                         
   - 4 A  C Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)]))

   + 2 A C 

                                                        2
  Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)])) 

          2 
   - 4 A C  

                                                        3
  Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)])) 

      2                                                       4
   + C  Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)])) 


;
G25 := G(xi) = -1/(2*C)*(sqrt(-2*A*C) + sqrt(-6*A*C)*tanh(1/2*sqrt(-6*A*C)*xi));
G25 := Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)])) = - 

          (1/2)           (1/2)     /1         (1/2)   \
  (-2 A C)      + (-6 A C)      tanh|- (-6 A C)      xi|
                                    \2                 /
  ------------------------------------------------------
                           2 C                          


(simplify(odetest(G25, ode2)) assuming (A < 0, 0 < C));

@janhardo  we have to change the ode when we change the ode we have to replace A,B,C changed to thus in eq4 and 5 but how satisfy i don't have idia

@janhardo  after equation 24 the reds one is not satisfy the equation i have to use equation 4-5 condition but i don't know how they use and satisfy the ode 

@janhardo  in equation 4 to 5 there is some substitution maybe they use that can you arrange the condition

@dharr  i got the problem with you explanation i didn't see that when i copy and paste this xi[0] from a mw file to another i don't know why this is happen and when i rewrite again this xi[0] by the same xi[0] it will fix it , but in my function i didn't saw that what  you mentioned thank you so much dear dctor

@janhardo i try this i did this before for another ode , thank you so much john

G15 := G(xi) = -(1/(xi__0 + xi) + B/2)/C

@janhardo 
 

Download case9-not_satisfy.mw

@janhardo i will check but i need some time to check

@janhardo  can you try other question of mine i needed to do for some other ode  i have other one which is so close to this can you find even just one of thus solution satisfy the ode? can you do by your Ai really i needed to satisfy odetest at least for one of them , is similar to this ode but totally different but have a trick

@janhardo  can you try to get thus be zero too becuase i have too make zero all of them  my error is start at here i mentioned in the file , can you undrestand what will happen to this phi  inside function there is must be another idea

@ecterrab  i didn't saw that  , now i got the solved the  problem, thank to you .

@mmcdara after spending more than 2 day and changing variable one by one i get the best shape of existence  plot, your trail is so good and design are emazing, i like your style plot but i have a lot of this long function which i have to plot them becuase of that i asked is make me angry when i spen this time for one plot 

plots-long_term_.mw

@dharr  is so better than before becuase i have a lot of that U(xi) i am looking for them but this index work very good thanks 

@acer it pretty working with this kind of factoring it do all factoring this is so good and i like it, i try to use both of your code to see which one is better the factoring is better work because the other code make fraction  

Download work-latex.mw