@tomleslie 

you can try

inputvariable = [b(t)], outputvariable = [a(t)]

hope that there is one of example of graph 

@tomleslie 

this is why I said that to try all combinations of a,b, and c for input and output by define one or more of them as input and output.

i expect to plot arrow vector field plot to show sink and source in the sphere.

but first I have to plot in the plane.

then I will try to plot on the sphere 

next is use arrow field plot on sphere and expand it on a plane when means to use

plane to show the arrow situation on sphere.

@tomleslie 

actually there is no input and ouput in the system,

so there is a need to try all combinations of input and ouptut set

you can define any kind of input and output set,

hope to see the direction arrow in the grarph to show sink and source.

@Kitonum 

There is a theorem stated that there must be a zero if vector field on sphere.

is there an example to show that there is no zero in plane originally, but draw on sphere then there is zero?

would like to see this interesting example.

@Preben Alsholm 

i will post photo link here when i see this issue next time.

since i do not know when it occur, it usually depends on the worksheet,

i rememeber that i meet ordering change when it change in the middle time of using.

i discover this when i can not dsolve or dsolve result is different but the code is the same time i use last time.

@vv 

i think minimize algorithm should independent of the content of procedure

objectiveproc := proc(mmm)
local i, t, eignvalues1, InputMatrix3, InputMatrix3AA, BeforeOld_Asso_eigenvectorAA, MA1, MA2, MA3, s1, s2, s3, Old_Asso_eigenvectorM:
i := 0:
t := 1:
InputMatrix3 := Matrix([[mmm, close3(t+1+i) , close3(t+2+i)],
[close3(t+1+i) , close3(t+2+i),0],
[close3(t+2+i),0 , 0]]):
InputMatrix3AA := MatrixMatrixMultiply(Transpose(InputMatrix3), InputMatrix3);
InputMatrix3AA := InputMatrix3;
BeforeOld_Asso_eigenvectorAA := InputMatrix3AA - lambda*IdentityMatrix(3);
eignvalues1 := evalf(solve(Determinant(BeforeOld_Asso_eigenvectorAA), lambda));
MA1 := InputMatrix3AA - eignvalues1[1]*IdentityMatrix(3):
MA2 := InputMatrix3AA - eignvalues1[2]*IdentityMatrix(3):
MA3 := InputMatrix3AA - eignvalues1[3]*IdentityMatrix(3):
#reducedform(MA1);
#reducedform(MA2);
#reducedform(MA3);
s1 := NullSpace(MA1)[1];
s2 := NullSpace(MA2)[1];
s3 := NullSpace(MA3)[1];
Old_Asso_eigenvectorM := Matrix([[s1[1],s2[1],s3[1]],[s1[2],s2[2],s3[2]],[s1[3],s2[3],s3[3]]]);
return Trace(abs(Old_Asso_eigenvectorM));
end proc:

@tomleslie 

i solved again , discover diff(a(t),t) is not zero

diff(a(t), t) = diff(a(t), t);
diff(b(t),t) = 0;
diff(c(t),t) = -b(t)/a(t);

@Preben Alsholm 

sorry i solved again, it return diff(a(t), t) = diff(a(t), t), not diff(a(t), t) = 0

my previous question about how to find gradient,

diff(a(t), t) = diff(a(t), t)

i do not know a(t), but from diff(a(t),t) = 0, i know it is constant.

i feel odd about this system i created.

if a(t) and b(t) are constant, why diff(c(t),t) not equal to 0?

is it reasonable for this system exist?

i google tutorial , it express diff(a(t),t) and diff(b(t), t) and diff(c(t),t) in terms of a(t) and b(t) and c(t)

perhaps my invented system should not use equal to 0 when solve for the expression

@vv 

i solved it by subs(bt = b(t), subs(at = a(t), diff(subs(b(t) = bt, subs(a(t) = at,  -b(t)/a(t))), at)

i just wonder why it do not see a(t) as a variable to diff a(t)

@Adri van der Meer 

just build equation like Lego from determinant, my example include redundant terms

perhaps my determinant conjecture is wrong, however, it is near my result.

because it can not be factorized.

@Markiyan Hirnyk 

just for looking pretty from determinant of matrix

@tomleslie 

i run this file , got error in maple 18

i am wondering how this classic math book can be solved

i understand your comment , you said that u1(t) and u2(t) have already been solutions

but i still do not understand how different orders of differential terms be added and result in zero.

Error, (in PDEtools/NumerDenom) invalid input: `PDEtools/NumerDenom` expects its 1st argument, ee, to be of type algebraic, but received [[u1(t), -t^2*(-2*t*(diff(u1(t), t))-t^2*(diff(diff(u1(t), t), t)))], [u2(t), -t^2*(-2*t*(diff(u2(t), t))-t^2*(diff(diff(u2(t), t), t)))]]
Error, (in PDEtools/NumerDenom) invalid input: `PDEtools/NumerDenom` expects its 1st argument, ee, to be of type algebraic, but received [[u1(t), -t^2*(-2*t*(diff(u1(t), t))-t^2*(diff(diff(u1(t), t), t)))], [u2(t), -t^2*(-2*t*(diff(u2(t), t))-t^2*(diff(diff(u2(t), t), t)))]]

@tomleslie 

in fact, a book mention p and q depend on linearly independent solution u1 and u2 of this differential equation

Here are the definition of p and q, how to calculate in maple?

Fuchsian equations by Masaaki yoshida, from page 14 to page 19, p and q at page 19

f := diff(u(t), t$2) + mm1*diff(u(t),t) + mm0*u(t) = 0;

f := diff(u(t), t$2) + (2/t-1/t^2*p(1/t))*diff(u(t),t) + 1/t^4*q(1/t)*u(t) = 0;
dsol2 := dsolve(f,u(t));

p := t -> -1*Determinant(Matrix([[dsol2[1], diff(dsol2[1], t$2)],[dsol2[2], diff(dsol2[2],t$2)]]))/Determinant(Matrix([[dsol2[1], diff(dsol2[1], t)],[dsol2[2], diff(dsol2[2],t)]]));
q := t -> Determinant(Matrix([[diff(dsol2[1],t), diff(dsol2[1], t$2)],[diff(dsol2[2],t), diff(dsol2[2],t$2)]]))/Determinant(Matrix([[dsol2[1], diff(dsol2[1], t)],[dsol2[2], diff(dsol2[2],t)]]));

mm1 := 2/t-1/t^2*p(1/t);
mm0 := 1/t^4*q(1/t);

@JohnS 

i am am finding the rest of two roots on the line y=-z

and assume two data points are roots and the data is a polynomial intercept y=-z

actually above code has little incorrect, It should interpolate two times,

first time intercept y=-z first , second time use first time result' s points and interpolate again

@JohnS 

how to use NLPSolve in this case

> NLPSolve(sys1 = 0, {sys2 = 0}, assume = nonnegative);
Error, (in Optimization:-NLPSolve) objective function must be specified as a procedure or algebraic expression
> NLPSolve({sys1 = 0, sys2 = 0}, assume = nonnegative);
Error, (in Optimization:-NLPSolve) objective function must be specified as a procedure or algebraic expression