Finding the solution of this long equation using Maple

[jamesbrazil]

TL;DR

Command to solve a strange equation in Maple for $x \ll 1$

I need help solving an equation. I started using Maple, but had no success. Could someone explain to me which command to use? I need to find a very small value of $x$, that is, $x \ll 1$. The equation is

$$434972871000000000.0+{\frac {\sqrt {6} \left( { 1.488388992\times 10^
{-36}}\,\ln \left( 1/12\,{\frac {12\,\sqrt {6}{x}^{2}+{
1.488388992\times 10^{-36}}\,\sqrt {6}+12\,\sqrt {6\,{x}^{2}+{
1.488388992\times 10^{-36}}}x}{x}} \right) \sqrt {6}-12\,
\sqrt {6\,{x}^{2}+{ 1.488388992\times 10^{-36}}}x \right) }{
72\,{x}^{3}}}-{\frac { \left( 0.001704000000\,{x}^{2}+{
1.488388992\times 10^{-36}} \right) \left( { 1.488388992\times 10^{
-36}}\,\ln \left( 1/12\,{\frac { 0.003408000000\,\sqrt {6}{x}^{2
}+{ 1.488388992\times 10^{-36}}\,\sqrt {6}+12\,\sqrt {
0.0000004839360000\,{x}^{2}+{ 4.227024737\times 10^{-40}}}x
}{x}} \right) \sqrt {6}-12\,\sqrt { 0.0000004839360000\,{x}^
{2}+{ 4.227024737\times 10^{-40}}}x \right) \sqrt {6}}{72\,\sqrt
{ 6.000000001\,{x}^{2}+{ 5.240806311\times 10^{-33}}}\sqrt {
0.0000004839360000\,{x}^{2}+{ 4.227024737\times 10^{-40}}}{
x}^{3}}}=0$$

 

[renormalize]

TL;DR Summary: Command to solve a strange equation in Maple for $x \ll 1$

I need help solving an equation. I started using Maple, but had no success. Could someone explain to me which command to use? I need to find a very small value of $x$, that is, $x \ll 1$. The equation is

$$434972871000000000.0+{\frac {\sqrt {6} \left( { 1.488388992\times 10^
{-36}}\,\ln \left( 1/12\,{\frac {12\,\sqrt {6}{x}^{2}+{
1.488388992\times 10^{-36}}\,\sqrt {6}+12\,\sqrt {6\,{x}^{2}+{
1.488388992\times 10^{-36}}}x}{x}} \right) \sqrt {6}-12\,
\sqrt {6\,{x}^{2}+{ 1.488388992\times 10^{-36}}}x \right) }{
72\,{x}^{3}}}-{\frac { \left( 0.001704000000\,{x}^{2}+{
1.488388992\times 10^{-36}} \right) \left( { 1.488388992\times 10^{
-36}}\,\ln \left( 1/12\,{\frac { 0.003408000000\,\sqrt {6}{x}^{2
}+{ 1.488388992\times 10^{-36}}\,\sqrt {6}+12\,\sqrt {
0.0000004839360000\,{x}^{2}+{ 4.227024737\times 10^{-40}}}x
}{x}} \right) \sqrt {6}-12\,\sqrt { 0.0000004839360000\,{x}^
{2}+{ 4.227024737\times 10^{-40}}}x \right) \sqrt {6}}{72\,\sqrt
{ 6.000000001\,{x}^{2}+{ 5.240806311\times 10^{-33}}}\sqrt {
0.0000004839360000\,{x}^{2}+{ 4.227024737\times 10^{-40}}}{
x}^{3}}}=0$$

Can you repost this equation, using symbols for all of the lengthy decimal numbers, followed with a separate list of all the symbols and their respective decimal values? That would make everything much easier to read and work with!

 

[Bill Simpson]

If anyone might still be interested in this then I've tried to reformat it as requested by jamesbrazil

a=434972871000000000.0;
b=1.488388992*10^-36;
c=0.001704000000;
d=0.003408000000;
e=0.0000004839360000;
f=4.227024737*10^-40;
g=5.240806311*10^-33;
a+
(sqrt(6)*(b*ln(1/12*
12*sqrt(6)*x^2+b*sqrt(6)+
12*sqrt(6*x^2+b)*x)/x))*sqrt(6)-
12*sqrt(6*x^2+b)*x)/(72*x^3)-
((c*x^2+b)*
(b*ln(1/12*(d*sqrt(6)*x^2+b*sqrt(6)+
12*sqrt(e*x^2+f)*x)/x)*sqrt(6)-
12*sqrt(e*x^2+f)*x)*sqrt(6))/
(72*sqrt(6*x^2+g)*
sqrt(e*x^2+f)*x^3)

I believe there is a root for x<<1 that lies between
8.071385266*10^-19 and 8.071385267*10^-19

It is certainly possible that I've made a mistake when trying to reformat this.
And I suppose it might be possible that the OP made a mistake when trying to format this.

Please check this carefully to try to find any of my mistakes before depending on this.

When I look at a plot of this it seems that the expression is approximately 4.34973*10^17
for a range of modest positive x.
It also seems that it is approximately that same value for many small negative x values,
but it seems there may be a number of points which are indeterminant.

 

[topsquark]

TL;DR Summary: Command to solve a strange equation in Maple for $x \ll 1$

I need help solving an equation. I started using Maple, but had no success. Could someone explain to me which command to use? I need to find a very small value of $x$, that is, $x \ll 1$. The equation is

$$434972871000000000.0+{\frac {\sqrt {6} \left( { 1.488388992\times 10^
{-36}}\,\ln \left( 1/12\,{\frac {12\,\sqrt {6}{x}^{2}+{
1.488388992\times 10^{-36}}\,\sqrt {6}+12\,\sqrt {6\,{x}^{2}+{
1.488388992\times 10^{-36}}}x}{x}} \right) \sqrt {6}-12\,
\sqrt {6\,{x}^{2}+{ 1.488388992\times 10^{-36}}}x \right) }{
72\,{x}^{3}}}-{\frac { \left( 0.001704000000\,{x}^{2}+{
1.488388992\times 10^{-36}} \right) \left( { 1.488388992\times 10^{
-36}}\,\ln \left( 1/12\,{\frac { 0.003408000000\,\sqrt {6}{x}^{2
}+{ 1.488388992\times 10^{-36}}\,\sqrt {6}+12\,\sqrt {
0.0000004839360000\,{x}^{2}+{ 4.227024737\times 10^{-40}}}x
}{x}} \right) \sqrt {6}-12\,\sqrt { 0.0000004839360000\,{x}^
{2}+{ 4.227024737\times 10^{-40}}}x \right) \sqrt {6}}{72\,\sqrt
{ 6.000000001\,{x}^{2}+{ 5.240806311\times 10^{-33}}}\sqrt {
0.0000004839360000\,{x}^{2}+{ 4.227024737\times 10^{-40}}}{
x}^{3}}}=0$$

If you know that x is small, I'd suggest a Taylor expansion first (use mtaylor in Maple), then solve it. That way it's just a polynomial equation.

-Dan

 

[Bill Simpson]

If anyone might still be interested in this then I've tried to reformat it as requested by jamesbrazil.

In experimenting with that I see that I dropped a couple of ( ) in my post above.

I apologize for that and I can't seem to edit that now to correct those mistakes.

This will hopefully correct my ( ) mistakes.

a=434972871000000000.0;
b=1.488388992*10^-36;
c=0.001704000000;
d=0.003408000000;
e=0.0000004839360000;
f=4.227024737*10^-40;
g=5.240806311*10^-33;
a+
(sqrt(6)*(b*ln(1/12*
(12*sqrt(6)*x^2+b*sqrt(6)+
12*sqrt(6*x^2+b)*x)/x))*sqrt(6)-
12*sqrt(6*x^2+b)*x)/(72*x^3)-
((c*x^2+b)*
(b*ln(1/12*(d*sqrt(6)*x^2+b*sqrt(6)+
12*sqrt(e*x^2+f)*x)/x)*sqrt(6)-
12*sqrt(e*x^2+f)*x)*sqrt(6))/
(72*sqrt(6*x^2+g)*
sqrt(e*x^2+f)*x^3)

Strangely enough, I now believe there may be two roots for x<<1

One may lie between 10^-37 and 10^-36 and may lie near 3.0381613366*10^-37

One may lie between 10^-28 and 10^-27 and may lie near 6.8635202356*10^-28

But with the size of the numbers involved this problem is very delicate and needs some care.

If anyone wants to try to check this then I would urge you to begin with checking that I have reformatted the original expression correctly and then begin carefully looking for the roots. I would greatly appreciate if someone would do that independently, I am always happy when someone finds and points out any of my mistakes.

Thank you and again I apologize for any and all of my errors.

 

[renormalize]

If anyone might still be interested in this then I've tried to reformat it as requested by jamesbrazil.

Frankly, until @jamesbrazil returns to show that he's still interested (and to explain the problem background and show the derivation his equation) I'm don't think this is worth pursuing.