# Question:obtaining the maximum value of a variable after using dsolve

## Question:obtaining the maximum value of a variable after using dsolve

Dear all,

I have a question regarding the dsolve procedure in Maple. I'm trying to construct a neutron star model using the Tolman-Oppenheimer-Volkoff equation using a polytropic equation of state (EOS) which requires me to solve the ode system:

diff(rho(r),r)=-(rho(r)*(1+K/(5/3-1)*rho(r)^(5/3-1))+K*rho(r)^(5/3))*(4*Pi*r^3*K*rho(r)^(5/3)+m(r))/(K*5/3*rho(r)^(5/3-1)*r*(r-2*m(r)))

diff(m(r),r)=4*Pi*rho(r)*(1+K/(5/3-1)*rho(r)^(5/3-1)

where I have used the EOS P(r)=K*rho(r)^(5/3) and K is a known constant. rho is the density of the star, m it's mass and P the pressure inside the star.

For the initial conditions I have chosen: rho(10^(-10))=rho_0 and m(10^-10)=0. I have chosen r=10^-10 as the innermost point for the integration, since the differential equation for rho is singular at r=0. rho_0 is the central density of the star.

I solve these equations numerically using:

TOV:=dsolve({ode,ics},numeric,output=listprocedure)

where ode is my system of differential equations and ics are my initial conditions. I need now the radius of the star (R_star), which is the maximum value of r, up until which Maple has carried out the integration.

My problem is, I don't know of any efficient way, to do this. What I'm doing currently is defining a procedure TOVr:=rhs(TOV[1]) and I evaluate it at a very high value of r, for which Maple returns me the error message: "Error, (in TOVr) cannot evaluate the solution further right of ..., probably a singularity". I then use the command TOVr('last') to call the maximum value of r and to store it.

I can use the above method, as long as I'm solving the ODEs only for a few different values of rho_0. But I would like to plot m(R_star) for values of rho_0 ranging from 10^(-14) to 10^(-12) in order to find the value of rho_0, for which I can obtain the maximum value for m(R_star). But this requires me to know the value of R_star for every rho_0 and using the above method is not feasible for say hundred different values of rho_0, since I can't write a loop, because it get's terminated as soon as Maple gives me the first error message.

I was thinking of using perhaps the 'events' command in dsolve, to stop the numeric integration once the value for the pressure drops very low, say below 10^(-46), since the radius at which P(r)=0 defines the stellar surface. I tried using:

TOV:=dsolve({ode,ics},numeric, events=[[K*rho(r)^(5/3)-10^(-46),halt]])

but if I try again to evaluate the solution at a large value of r, I get the above error message, and the integration doesn't get canceled, although the value 10^(-46) is bigger than the value for the pressure I would obtain for R_star using TOVr('last') and Maple shouldn't encounter a singularity.

Am I using the 'events' command wrong? And does somebody know of a more efficient method to obtain the maximum value of a variable after carying out a numerical integration using dsolve?

Sorry for the long post and thank you all.

﻿