Hi Alexander
1.
One of the tools available is MuPAD, start MuPAD with
once open define the equation, but avoid the abs(), it only complicates the result and since r seems to be a cylindrical coordinate, if r is the radius, then there is no point that r takes negative values
.
the result after ode:solve(o) is what comes up if including the abs()
the resulting functions do not seem that useful.
2.
It turns out that the Jacobi elliptic functions are solution to a similar differential equation:
clear all;clc;close all
rspan=[-20 20];
dr=0.0001;
r=[rspan(1):dr:rspan(2)];
M = 0.7;
[S,C,D] = ellipj(r,M);
figure(1);plot(r,S,r,C,r,D);
legend('SN','CN','DN','Location','best');grid on;
title('Jacobi Elliptic Functions sn,cn,dn');
a=1;b=1;c=.1;d=.1;
vS=d2ydr2(r,S,a,b,c,d);
vC=d2ydr2(r,C,a,b,c,d);
vD=d2ydr2(r,D,a,b,c,d);
d2Sdt2=diff(diff(S));
d2Cdt2=diff(diff(C));
d2Ddt2=diff(diff(D));
errS=d2Sdt2-vS([3:end]);
errC=d2Cdt2-vC([3:end]);
errD=d2Ddt2-vD([3:end]);
figure(2);plot(r([3:end]),abs(errS),r([3:end]),abs(errC),r([3:end]),abs(errD));grid on;
3.
now M=.99 CN and DN (not the blue one) show far smaller error
rspan=[1 5];
dr=0.0001;
r=[rspan(1):dr:rspan(2)];
M = 0.99;
[S,C,D] = ellipj(r,M);
vS=d2ydr2(r,S,a,b,c,d);
vC=d2ydr2(r,C,a,b,c,d);
vD=d2ydr2(r,D,a,b,c,d);
d2Sdt2=diff(diff(S));
d2Cdt2=diff(diff(C));
d2Ddt2=diff(diff(D));
errS=d2Sdt2-vS([3:end]);
errC=d2Cdt2-vC([3:end]);
errD=d2Ddt2-vD([3:end]);
figure(3);plot(r([3:end]),abs(errS),r([3:end]),abs(errC),r([3:end]),abs(errD));grid on;
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thanks in advance for time and attention
John BG
