@Adri van der Meer 

x1 := x/y;
x2 := x/y;
T := lexdeg([x1,x2],[e1,e2]);
GB := Basis([e1-x1,e2-x2],T);

Normal(-2*y/x, GB, T);

failed in this case

@Adri van der Meer 

any general method to find test4

is there a book teaching this

@Adri van der Meer 

what are the steps of this general method to find -(x^2+x*y)/2 ?

as i would like to apply this in general case

@Preben Alsholm 

after multiply 10000

eqt := (y^5+y^4 + -0.5109*y^3 + -0.0595*y^2 + 0.0000-(u^3 + -0.5109*u^2 + -0.0595*u))*10000 = 0;

> eqt := 10000*y^5+10000*y^4-5109*y^3-595*y^2-10000*u^3+5109*u^2+595*u = 0*algcurves[parametrization](eqt, y, u, t);
Error, (in algcurves/parametrization) Input should be a polynomial in, y, u

other methods do not have result

The exact i want to do is this example

https://docs.google.com/file/d/0Bxs_ao6uuBDUT25BeVhtWjZwRUU/edit?usp=sharing
https://docs.google.com/file/d/0Bxs_ao6uuBDUVXU5dXh6Rl9zUUU/edit?usp=sharing

are you here? do you see this message?

The exact i want to do is this example

https://docs.google.com/file/d/0Bxs_ao6uuBDUT25BeVhtWjZwRUU/edit?usp=sharing
https://docs.google.com/file/d/0Bxs_ao6uuBDUVXU5dXh6Rl9zUUU/edit?usp=sharing

are you here? do you see this message?

https://docs.google.com/file/d/0Bxs_ao6uuBDUT25BeVhtWjZwRUU/edit?usp=sharing
https://docs.google.com/file/d/0Bxs_ao6uuBDUVXU5dXh6Rl9zUUU/edit?usp=sharing

Question 2 is from this example

actually this example has 3 steps

step 2 is this question 1

step 1 and step 3 is Question 2

https://docs.google.com/file/d/0Bxs_ao6uuBDUT25BeVhtWjZwRUU/edit?usp=sharing
https://docs.google.com/file/d/0Bxs_ao6uuBDUVXU5dXh6Rl9zUUU/edit?usp=sharing

Question 2 is from this example

actually this example has 3 steps

step 2 is this question 1

step 1 and step 3 is Question 2

i just want to know how to multiply and integrate and get

x1 -> Diff(lambda1(t),t) - lambda3(t) = miu1(t)

x2 -> Diff(lambda2(t),t) + lambda1(t) - 4*lambda3(t) = miu2(t)

x3 -> Diff(lambda3(t),t) + lambda2(t) - 3*lambda3(t) = miu3(t)

u  -> 2*Diff(lambda3(t),t$2) - 5*Diff(lambda3(t),t) + 7*lambda3(t) = miu(t)

is it

-lambda1(t)*(Diff(x1(t),t) - x2(t)) = 0;

-lambda2(t)*(Diff(x2(t),t) - x3(t)) = 0;

-lambda3(t)*(Diff(x3(t),t) +3*x3(t) + 4*x2(t) + x1(t)) = -lambda3(t)*(2*Diff(u(t),t$2) + 5*Diff(u(t),t) + 7*u(t));

i can not see how integration get result above, where do miu1(t) come from?

can solve do this? if not, how to do?

i see the manual of Involutive or Ore, it said R adjoint is involutive

if it is correct, how to make a command to use involutive

i just want to know how to multiply and integrate and get

x1 -> Diff(lambda1(t),t) - lambda3(t) = miu1(t)

x2 -> Diff(lambda2(t),t) + lambda1(t) - 4*lambda3(t) = miu2(t)

x3 -> Diff(lambda3(t),t) + lambda2(t) - 3*lambda3(t) = miu3(t)

u  -> 2*Diff(lambda3(t),t$2) - 5*Diff(lambda3(t),t) + 7*lambda3(t) = miu(t)

is it

-lambda1(t)*(Diff(x1(t),t) - x2(t)) = 0;

-lambda2(t)*(Diff(x2(t),t) - x3(t)) = 0;

-lambda3(t)*(Diff(x3(t),t) +3*x3(t) + 4*x2(t) + x1(t)) = -lambda3(t)*(2*Diff(u(t),t$2) + 5*Diff(u(t),t) + 7*u(t));

i can not see how integration get result above, where do miu1(t) come from?

can solve do this? if not, how to do?

i see the manual of Involutive or Ore, it said R adjoint is involutive

if it is correct, how to make a command to use involutive

how to solve this system, i increase the number of equations from 6 to 8 still, get infinite solutions

if this method not work, can least square method work, how to apply in this case?

eqt := Matrix(8,1);
for i from 1 to 8 do
#a*y' + b*y'' + c*y - (d*u' + e*u'' + g*u) = 0;
eqt[i,1] := (a*(x[i]-x[i+2]) + b*(x[i]-2*x[i+1]+x[i+2])/2 + c*x[i] - (d*(x[i+1]-x[i+3]) + e*(x[i+1]-2*x[i+2]+x[i+3])/2 + g*x[i+1]))[1] = 0;
od;

a*y' + b*y'' + c*y - (d*u' + e*u'' + g*u) = 0;

sys := [
1.8*a+.4000000000*b+103.9*c-.5000000000*e-102.6*g = 0,
.5000000000*b+102.6*c+.3*d+.3500000000*e-102.1*g = 0,
-3.3*a-1.050000000*b+102.4*c+.7*d+.2500000000*e-105.1*g = 0,
-2.5*a+1.450000000*b+102.6*c+3.3*d+1.050000000*e-102.4*g = 0,
-1.6*a+1.100000000*b+105.8*c+.3*d+1.750000000*e-105.5*g = 0,
-.7*a-.2500000000*b+105.1*c-.2*d+.2000000000*e-105.7*g = 0,
-.3*a-.3500000000*b+102.1*c+2.5*d-1.450000000*e-102.6*g = 0,
.2*a-.2000000000*b+105.7*c+1.6*d-1.100000000*e-105.8*g = 0];
sol := solve(sys);
eval(sys, sol);

how to solve this system, i increase the number of equations from 6 to 8 still, get infinite solutions

if this method not work, can least square method work, how to apply in this case?

eqt := Matrix(8,1);
for i from 1 to 8 do
#a*y' + b*y'' + c*y - (d*u' + e*u'' + g*u) = 0;
eqt[i,1] := (a*(x[i]-x[i+2]) + b*(x[i]-2*x[i+1]+x[i+2])/2 + c*x[i] - (d*(x[i+1]-x[i+3]) + e*(x[i+1]-2*x[i+2]+x[i+3])/2 + g*x[i+1]))[1] = 0;
od;

a*y' + b*y'' + c*y - (d*u' + e*u'' + g*u) = 0;

sys := [
1.8*a+.4000000000*b+103.9*c-.5000000000*e-102.6*g = 0,
.5000000000*b+102.6*c+.3*d+.3500000000*e-102.1*g = 0,
-3.3*a-1.050000000*b+102.4*c+.7*d+.2500000000*e-105.1*g = 0,
-2.5*a+1.450000000*b+102.6*c+3.3*d+1.050000000*e-102.4*g = 0,
-1.6*a+1.100000000*b+105.8*c+.3*d+1.750000000*e-105.5*g = 0,
-.7*a-.2500000000*b+105.1*c-.2*d+.2000000000*e-105.7*g = 0,
-.3*a-.3500000000*b+102.1*c+2.5*d-1.450000000*e-102.6*g = 0,
.2*a-.2000000000*b+105.7*c+1.6*d-1.100000000*e-105.8*g = 0];
sol := solve(sys);
eval(sys, sol);

progress is

i separate constant into another equation

eliminate([x0 = 3*u+2*v+w, x1 = 4*u+3*v+w, x0 + x1 = 22],{u,v,w},'s');

u+v = -x0+x1

is it result = -x0+x1 = 22

if possible, i can change system of equation into surface

progress is

i separate constant into another equation

eliminate([x0 = 3*u+2*v+w, x1 = 4*u+3*v+w, x0 + x1 = 22],{u,v,w},'s');

u+v = -x0+x1

is it result = -x0+x1 = 22

if possible, i can change system of equation into surface

sorry for misunderstanding, the goal is to convert parametric equation back to non-parametric equation, only contains x0 and x1