i fixed just be remove one term in (eqw) remove the term which have k[i] it will fixed and make the pde to zero 

@dharr you are right about that, it is a function of t  i don't have idea the only way for knowing that is that pdes make zero for other function  but in here i  get  error in pdes test  i don't know how work on it , and also if we change the function to sin(t) and cos(t) and even t must be work , but i get exccet 10000000

tester2.mw 

@dharr my bad, i am looking for find R somehow be a number or at least be a function of t, i write two pdes which one of them pde1 and second one i did pde=sub(function of t=#) into  pde1, i saw in a lot of pdes which they solved like that when they change the function of t to anotehr function or just equal them to 1 after that the we can find that R in R*ln(f)_x otherwise we can't i have a lot example you can tested without that change you can't find that R, i want arrange of that by changing the function of t to another function  and when i found R i will say under this condition we can solved the pdes this kind of solving have a different of plots  i  want somehow find that condition by changing function of t becuase there is a lot of pdes which not solved yet, if i do that it will be a great job but i don't know how many condition i have to do, i will put some picture with mw file and see after changing function of t then we can find that R, i have a lot paper they did but i just put some of them ....

the pictuer and mw file are respectivly ...

p1.mw

p2.mw

p3.mw

in p1 and p3 i found that changing in pde=subs() and in p2 as mention in picture two the author found it, there is a lot of pde not founded yet, how we can find that changing of function to find that R ? if chang one of them you can't find that R test?

any other question please mention

@acer he already find it but is wrong, i already use the 6/beta*ln(f)_x which is R*ln(f)_x  i have too find this R, but in a lot paper we have to do something to our pde by constract some conditions by function of t except makeing them to zero then we can finded, in the first paper i finded when i did the author changind like he did in first picture equation(1)  but in second paper i can't find that transformation R ln(f)_x like  author did and i already use the existing function is not work, and my procedure is work for %99 of pdes, i think is my openion i have to do something with pdes to find that R ln(f)_x this R is neccesary ...

this is the first equation i founded but without that change we can't finded 

k.mw

i want to do the same for  the picture two which contain that equation (1)

@Ronan thanks a lot, i change the equatoion becuase that issue and now i am back by your code i fixed thanks a lot ...

@janhardo not work always put mw file 

@sand15 by this pretty code we know which function we can use so, how i can add more function like exponential and even polynomial function like t, t^2 and so one and how many trigonometric and hyperbolic functiona ll of them i want all of them , beside this yesterday i solved another function which contain i  think 5 different function like g(t),p(t) q(t) and so on i will put that function in here too how we can do some iteration work for them  between them for all that function, the code is so pretty if possible add more function for the code also for generate second code is hard is not easy but maybe possible is just idea 

t-sand-mccdara.mw

one author use thus function just how i can test other  all of function i want the design and shape all of them to see which one i can put inside the explore option, like that p(t)=t, q(t)=cos(t), chi(t)=sinh and so one some interaction betwen them what we have if be like that

@dharr you are right about this ode, but the strange thing is this when i used for making my  general ode be zero is worked, i am in shock  with this i don't know use the function or not becuase is not solution of the auxiliary ode but is solution for general ode. i don't know what to do, please give me advise about that?

and i make it short the steps and remove some steps in case not be so long.

problem-.mw

@janhardo is like that i finished that 

line-3-done.mw

@janhardo  but i think  this plot is belong to mathematica program i didn't search but i have to plot different i am keep looking and thinking for get new idea which not repeated

@janhardo  i have to change my design of graph , matlab not so suir for thus kind of plotting 

something like that

@sand15 do you think i can do that and i don't want,  my research is stuck becuase of this step but at end with your help i finish the work this time is work and i construct 3 line thank to you,  i can't see the error like you do, i did watch and did some pertubation but didn't work in my hand 

@janhardo  they must transform for complex number 

design of graph my means? how many design we can have it for plots of lump even in 3D like this function 

2*((t*(-alpha*conjugate(lambda[1] + lambda[2]*I)^3 + b*conjugate(lambda[1] + lambda[2]*I) + c*r[2] + a) + 2*beta*(y*conjugate(lambda[1] + lambda[2]*I) + z*r[2] + x))/(2*beta) + (t*(-alpha*(lambda[1] + lambda[2]*I)^3 + b*(lambda[1] + lambda[2]*I) + c*r[1] + a) + 2*beta*(y*(lambda[1] + lambda[2]*I) + z*r[1] + x))/(2*beta))/(((t*(-alpha*(lambda[1] + lambda[2]*I)^3 + b*(lambda[1] + lambda[2]*I) + c*r[1] + a) + 2*beta*(y*(lambda[1] + lambda[2]*I) + z*r[1] + x))*(t*(-alpha*conjugate(lambda[1] + lambda[2]*I)^3 + b*conjugate(lambda[1] + lambda[2]*I) + c*r[2] + a) + 2*beta*(y*conjugate(lambda[1] + lambda[2]*I) + z*r[2] + x)))/(4*beta^2) - 4/((lambda[1] + lambda[2]*I - conjugate(lambda[1] + lambda[2]*I))^2*alpha))

can you plot by Ai you have lets see how many shape you have ?

@sand15 it is good to have you in here mapleprime need you...
i see yourse plot in below i run but didn't work i try to copy your function in my file again give me another error is nonsense a lot what the different betwen yours and mine? is look same 

Line-1-Done.mw

@janhardo  this is a little bit complecated, but you plotting in last is so pretty how many graph and hsape of design you can make it?