Handling constant names for ODE solutions

So, I try to solve symbolycally a variational problen of finding minima for a given functional. But I can't handle names of constants while solving Euler's equation. when changing function F, constants enumerate differently and i don't know how to keep them in place/

clc 
syms x y Dy D2y 
F= x + y^2 + Dy^2;
x1=0;
y1=1;
x2=1;
y2=3;
phi= -x^2 + 4; 

fprintf('F(x,y,y'')=%s\n',char(F))
fprintf('Boundary cond. on left  side: y(%d)=%d\n',x1,y1)
fprintf('Boundary cond. on right side: y(%d)=%d\n',x2,y2)
fprintf('Right side restrictions: y=%s\n',char(phi))
dFdy=diff(F,y); 
dFdy1=diff(F,Dy); 
d_dFdy1_dx=diff(dFdy1,x); 
d_dFdy1_dy=diff(dFdy1,y); 
d_dFdy1_dy1=diff(dFdy1,Dy); 
dFy1dx=d_dFdy1_dx+d_dFdy1_dy*Dy+d_dFdy1_dy1*D2y;

Euler=simplify(dFdy-dFy1dx); 
deqEuler=[char(Euler) '=0']; 
Sol=dsolve(deqEuler,'x'); 
if length(Sol)~=1 
  error('No solutions (or more then 1)!');
end
SolLeft=subs(Sol,x,x1); 
SolRight=subs(Sol,x,x2); 

EqLeft=[SolLeft == sym(y1)] % =y1
EqRight=[SolRight == sym(y2)]; % =y2
syms C1 C2

Con=solve([EqLeft EqRight],[C1 C2]);

So, plugging for example F = -y^2 + Dy^2 gives me system with C6 and C7 that pop out of nowhere instead of C1 and C2. Why is this so?

Answer 1

[Walter Roberson]

MATLAB tends to use the "next available" constant of integration -- but "next available" might take into account that the symbolic engine still has internal references to the C* that was used earlier.

You need to use symvars() or similar to detect which names are there, possibly with a setdiff() against the known names of symbols.

When there are multiple constants in the same expression, then sometimes you need to do a bit of analysis to figure out which constant is acting in which role. Sometimes it is easiest to substitute known values such as 0 or 1 for variables and see which constant surives. Sometimes, though, you need to use something like findSymType() to match a particular subexpression... it can get to be a bit of a nuisance.