I want to find some more thing in it.last time you help me in solving my equations and you wrote a command for the graphical relation between mass and density. unfortunately, it does not response me.I show you the right curve and want to find the graphical relation between pressure(P) and density(rho)... and the value of r (radius) where P touch become 0.

 right_graph.mw

it is not necessary that horizontal axis must start at rho=0(since rho=0 is not define). it may start with some suitable value.

@tomleslie could their be possible graph between P vs rho, and P vs epsilon.

@tomleslie u take density in first condition as rho<180.4,undefined, all right b/c below 180.4 density is not define.

in 2nd case u took density as:rho<378.65,alpha[0] rho+beta[0]^Gamma__0. it also right b/c in this case the rho(density) lies in the region of [180.4,378.65] and it is the1st define region for rho.

but in the third case u wrote  as:rho<605.84,alpha[1] rho+beta[1]^Gamma__1, as I understand the  meaning it says, for i=1 rho contain all value that are less than 605.84, but in this case we have to take densities that lie in the region [378.65,605.84], not less than 378.65.not any  region of density meet with the other region. each region is uniquely define in their respective interval of densities. same for the last rho<1060.74,alpha[2] rho+beta[2]^Gamma__2,and the in this region we define density in the interval [605.84,1060.74]. for this last third region,we don't need to asign densities that are less than 605.84.

there are three region for rho(density) and each region is defined in its own density interval and no region intersect with other one.

@Carl Love 

@tomleslie 

i have read before and now several times, carefully. but, may be its my lackness in understanding or weak grip on maple... and i could not understand the meaning of "NB" as in file you wrote

NB This will fail because the system is
# singular at the left end-point - pretty
# obvious from the denominators in the set
# 'odes' above. With x=0, three of the four
# RHS are infinite
 i know the solution  is not well defines at 0. therefore chosen eps:=1e-07.i accept that i didn't stated before that i am not interested in complex behaviour. 

i also have solved the begining three initial value problems... as you stated but how to obtained the solution for coupled with 4 that is problematic.the result may cantain huge values!

hope u understand

badODEsys.mw

@tomleslie 

the problem creator is the 4th one... don't you think so it can handle with midrich or midpoint code some...and i think midrich or midpoint can deal with singularity... we can use near to zero like 1e-07

i also tried to solve it by following your instruction i got error of missing boundary,,, i think it is sufficient 4 condiions for 1st order equations. 

odes.mw