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Question: Do I really get all values from this RootOf expression

Question:Do I really get all values from this RootOf expression

C_R 2607 Maple
rootof allvalues
July 06 2024
2

I was about to verify two solutions of dsolve from here but could not find an agreement for negative values. This makes me wonder if all values are computed.

There is also a different behaviour that I do not understand when allvalues is given a RootOf expression or an equation containing a RootOf expression.
 

dsolve without method

> 

ode:=diff(y(x), x) = (3*x - y(x) + 1)/(3*y(x) - x + 5);
ic:=y(0)=0;
dsolve({ode,ic});
plot(rhs(%),x=-10..10,numpoints=10);
evalf(subs(x=3,%%));
evalf(subs(x=-3,%%%));

diff(y(x), x) = (3*x-y(x)+1)/(3*y(x)-x+5)

  

y(0) = 0

  

y(x) = -(-(1/36)*(x+1)^2*((-324+12*(96*x^3+288*x^2+288*x+825)^(1/2))^(2/3)-24*x-24)^2/(-324+12*(96*x^3+288*x^2+288*x+825)^(1/2))^(2/3)-x^3-x^2+x+1)/(x+1)^2

  

 

y(3) = 2.135964164

  

y(-3) = -2.302775638+0.4883358175e-9*I

 (1)

dsolve with a particular method

> 

sol:=dsolve([ode,ic],[dAlembert]);
odetest(sol,[ode,ic]);

y(x) = RootOf(-6*((x+_Z+3)/(5+3*_Z-x))^(2/3)*(-4*(-x+_Z+1)/(5+3*_Z-x))^(1/3)*_Z+2*((x+_Z+3)/(5+3*_Z-x))^(2/3)*(-4*(-x+_Z+1)/(5+3*_Z-x))^(1/3)*x+2*(-4)^(1/3)*3^(2/3)-10*((x+_Z+3)/(5+3*_Z-x))^(2/3)*(-4*(-x+_Z+1)/(5+3*_Z-x))^(1/3))

  

[0, 0]

 (2)

Since allvalues fails on this expression for real valued x, rational and integer values are tried for punctual comaprision

> 

subs(x=3,sol);
allvalues(%);
evalf(%)

y(3) = RootOf(-6*((6+_Z)/(2+3*_Z))^(2/3)*(-4*(-2+_Z)/(2+3*_Z))^(1/3)*_Z-4*((6+_Z)/(2+3*_Z))^(2/3)*(-4*(-2+_Z)/(2+3*_Z))^(1/3)+2*(-4)^(1/3)*3^(2/3))

  

y(3) = (6873/256)*(-(4/6873)*(20994724+82476*6873^(1/2))^(1/3)-(128/3)/(20994724+82476*6873^(1/2))^(1/3)+1/3-((1/2)*I)*3^(1/2)*((8/6873)*(20994724+82476*6873^(1/2))^(1/3)-(256/3)/(20994724+82476*6873^(1/2))^(1/3)))^2+(1/96)*(20994724+82476*6873^(1/2))^(1/3)+(2291/3)/(20994724+82476*6873^(1/2))^(1/3)-8947/768+((2291/256)*I)*3^(1/2)*((8/6873)*(20994724+82476*6873^(1/2))^(1/3)-(256/3)/(20994724+82476*6873^(1/2))^(1/3)), y(3) = (6873/256)*((8/6873)*(20994724+82476*6873^(1/2))^(1/3)+(256/3)/(20994724+82476*6873^(1/2))^(1/3)+1/3)^2-(1/48)*(20994724+82476*6873^(1/2))^(1/3)-(4582/3)/(20994724+82476*6873^(1/2))^(1/3)-8947/768, y(3) = (6873/256)*((8/6873)*(20994724+82476*6873^(1/2))^(1/3)+(256/3)/(20994724+82476*6873^(1/2))^(1/3)+1/3)^2-(1/48)*(20994724+82476*6873^(1/2))^(1/3)-(4582/3)/(20994724+82476*6873^(1/2))^(1/3)-8947/768

  

y(3) = -6.067982077+1.049560864*I, y(3) = 2.13596417, y(3) = 2.13596417

 (3)

Two roots match the dsolve solution without method. However doing the same only on the right hand side produces different output. For some reason allvalues produces 3 RootOf expressions with a numerical root selector.

> 

subs(x=3,rhs(sol));
allvalues(%);
evalf(%)

RootOf(-6*((6+_Z)/(2+3*_Z))^(2/3)*(-4*(-2+_Z)/(2+3*_Z))^(1/3)*_Z-4*((6+_Z)/(2+3*_Z))^(2/3)*(-4*(-2+_Z)/(2+3*_Z))^(1/3)+2*(-4)^(1/3)*3^(2/3))

  

RootOf((3*I)*3^(1/6)*2^(2/3)+2^(2/3)*3^(2/3)-6*((6+_Z)/(2+3*_Z))^(2/3)*(-4*(-2+_Z)/(2+3*_Z))^(1/3)*_Z-4*((6+_Z)/(2+3*_Z))^(2/3)*(-4*(-2+_Z)/(2+3*_Z))^(1/3), -6.067982082+1.049560860*I), RootOf((3*I)*3^(1/6)*2^(2/3)+2^(2/3)*3^(2/3)-6*((6+_Z)/(2+3*_Z))^(2/3)*(-4*(-2+_Z)/(2+3*_Z))^(1/3)*_Z-4*((6+_Z)/(2+3*_Z))^(2/3)*(-4*(-2+_Z)/(2+3*_Z))^(1/3), 2.135964164), RootOf((3*I)*3^(1/6)*2^(2/3)+2^(2/3)*3^(2/3)-6*((6+_Z)/(2+3*_Z))^(2/3)*(-4*(-2+_Z)/(2+3*_Z))^(1/3)*_Z-4*((6+_Z)/(2+3*_Z))^(2/3)*(-4*(-2+_Z)/(2+3*_Z))^(1/3), 2.135964164-0.*I)

  

-6.067982082+1.049560860*I, 2.135964164, -6.067982082+1.049560860*I

 (4)

Why this change?
Now the same with a negative value. Now the root does not match the solution of the dsolve call without method.

> 

subs(x=-3,sol);
allvalues(%);
evalf(%)

y(-3) = RootOf(-6*(_Z/(8+3*_Z))^(2/3)*(-4*(4+_Z)/(8+3*_Z))^(1/3)*_Z-16*(_Z/(8+3*_Z))^(2/3)*(-4*(4+_Z)/(8+3*_Z))^(1/3)+2*(-4)^(1/3)*3^(2/3))

  

y(-3) = -1/2+(1/2)*13^(1/2)

  

y(-3) = 1.302775638

 (5)
> 

subs(x = -3, rhs(sol));
allvalues(%);
evalf(%);

RootOf(-6*(_Z/(8+3*_Z))^(2/3)*(-4*(4+_Z)/(8+3*_Z))^(1/3)*_Z-16*(_Z/(8+3*_Z))^(2/3)*(-4*(4+_Z)/(8+3*_Z))^(1/3)+2*(-4)^(1/3)*3^(2/3))

  

RootOf((3*I)*3^(1/6)*2^(2/3)-6*(_Z/(8+3*_Z))^(2/3)*(-4*(4+_Z)/(8+3*_Z))^(1/3)*_Z+2^(2/3)*3^(2/3)-16*(_Z/(8+3*_Z))^(2/3)*(-4*(4+_Z)/(8+3*_Z))^(1/3), 1.302775638+0.*I)

  

1.302775638

 (6)
> 

NULL

 

NULL


 

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