transcendental equation problem

[dji321]

i have a transcendental equation and i have not a mathematique superieur formation ( i'm an hydraulic engeneer) and i want to resolve it but i can't so if you can help me with it !!

the equation is : 2*x*n*ctg(2x)= x^2 - n^2 or (same equation) : (n*ctg(x)-x)*(n*tg(x)+x) =0

n= constante ( i have this n.... in example n = 0.5 )
xp = are the roots of equation above

---

and h_n is the root of the equation tan(2h) = \(\displaystyle \, \dfrac{2 \eta h}{h^2\, -\, \eta^2}\, \) or

\(\displaystyle \left(\eta \cot(h)\, -\, h\right)\left(\eta \tanh(h)\, +\, h\right)\, =\, 0\)

 

[Ishuda]

i have a transcendental equation and i have not a mathematique superieur formation ( i'm an hydraulic engeneer) and i want to resolve it but i can't so if you can help me with it !!

the equation is : 2*x*n*ctg(2x)= x^2 - n^2 or (same equation) : (n*ctg(x)-x)*(n*tg(x)+x) =0

n= constante ( i have this n.... in example n = 0.5 )
xp = are the roots of equation above

---

and h_n is the root of the equation tan(2h) = \(\displaystyle \, \dfrac{2 \eta h}{h^2\, -\, \eta^2}\, \) or

\(\displaystyle \left(\eta \cot(h)\, -\, h\right)\left(\eta \tanh(h)\, +\, h\right)\, =\, 0\)

I would think the easiest way to find the roots would be to find then individually, i.e., assuming the tanh is suppossed to actually be a tan, since
\(\displaystyle \left(\eta \cot(h)\, -\, h\right)\left(\eta \tan(h)\, +\, h\right)\, =\, 0\)
either
\(\displaystyle \eta \cot(h)\, -\, h\, =\, 0\)
or
\(\displaystyle \eta \tan(h)\, +\, h\, =\, 0\)
So find the zeros for each of the two simpler equations.

Now, since the range of tan(h) is \(\displaystyle (-\infty,\, \infty)\) on \(\displaystyle (-\frac{\pi}{2}\, +\, n\, \pi,\, \frac{\pi}{2}\, +\, n\, \pi)\), there will be a solution in each of those intervals for
\(\displaystyle \eta \tan(h)\, +\, h\, =\, 0\)
if \(\displaystyle \eta\, \ne\, 0\) and similarly for the equation with the cot.

EDIT: An obvious solution for the tan equation is zero for the interval n=0. For the interval n=1, h~1.8366

 

[dji321]

I would think the easiest way to find the roots would be to find then individually, i.e., assuming the tanh is suppossed to actually be a tan, since
\(\displaystyle \left(\eta \cot(h)\, -\, h\right)\left(\eta \tan(h)\, +\, h\right)\, =\, 0\)
either
\(\displaystyle \eta \cot(h)\, -\, h\, =\, 0\)
or
\(\displaystyle \eta \tan(h)\, +\, h\, =\, 0\)
So find the zeros for each of the two simpler equations.

Now, since the range of tan(h) is \(\displaystyle (-\infty,\, \infty)\) on \(\displaystyle (-\frac{\pi}{2}\, +\, n\, \pi,\, \frac{\pi}{2}\, +\, n\, \pi)\), there will be a solution in each of those intervals for
\(\displaystyle \eta \tan(h)\, +\, h\, =\, 0\)
if \(\displaystyle \eta\, \ne\, 0\) and similarly for the equation with the cot.

EDIT: An obvious solution for the tan equation is zero for the interval n=0. For the interval n=1, h~1.8366

i dont think this is the wright solution because it gives [FONT=MathJax_Main]± and infinity solution for just one term of n in my example where n=0.5 it gives this
[/FONT][FONT=MathJax_Math-italic]x[FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]7.91705268466621...[/FONT][/FONT]

[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]6.36162039206566...[/FONT][/FONT]

[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]4.81584231784594...[/FONT][/FONT]

[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]3.29231002128209...[/FONT][/FONT]

[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]1.83659720315213...[/FONT][/FONT]


[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]0.653271187094403 and i dont know how to solv it ,, i get this from my example and i need to solve it for many others n examples so please help me.[/FONT][/FONT]

 

[dji321]

I would think the easiest way to find the roots would be to find then individually, i.e., assuming the tanh is suppossed to actually be a tan, since
\(\displaystyle \left(\eta \cot(h)\, -\, h\right)\left(\eta \tan(h)\, +\, h\right)\, =\, 0\)
either
\(\displaystyle \eta \cot(h)\, -\, h\, =\, 0\)
or
\(\displaystyle \eta \tan(h)\, +\, h\, =\, 0\)
So find the zeros for each of the two simpler equations.

Now, since the range of tan(h) is \(\displaystyle (-\infty,\, \infty)\) on \(\displaystyle (-\frac{\pi}{2}\, +\, n\, \pi,\, \frac{\pi}{2}\, +\, n\, \pi)\), there will be a solution in each of those intervals for
\(\displaystyle \eta \tan(h)\, +\, h\, =\, 0\)
if \(\displaystyle \eta\, \ne\, 0\) and similarly for the equation with the cot.

EDIT: An obvious solution for the tan equation is zero for the interval n=0. For the interval n=1, h~1.8366

for n = 0.5 there is

[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]7.91705268466621...[/FONT]

[FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]6.36162039206566...[/FONT]

[FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]4.81584231784594...[/FONT]

[FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]3.29231002128209...[/FONT]

[FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]1.83659720315213...[/FONT]

[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]0.653271187094403...[/FONT]x[/FONT][FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]0.65327118709440[/FONT][/FONT]

 

[Ishuda]

i dont think this is the wright solution because it gives [FONT=MathJax_Main]± and infinity solution for just one term of n in my example where n=0.5 it gives this
[/FONT][FONT=MathJax_Math-italic]x[FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]7.91705268466621...[/FONT][/FONT]

[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]6.36162039206566...[/FONT][/FONT]

[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]4.81584231784594...[/FONT][/FONT][FONT=MathJax_Main]tan equation solution; j = 2[/FONT] for positive solution

[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]3.29231002128209...[FONT=MathJax_Main]cot equation solution[/FONT][FONT=MathJax_Main]
[/FONT][/FONT][/FONT]
[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]1.83659720315213...[FONT=MathJax_Main]tan equation solution; j =[/FONT][/FONT][/FONT][FONT=MathJax_Main][FONT=MathJax_Main] 1 f[/FONT][/FONT][FONT=MathJax_Main]or positive solution
[/FONT]
[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]≈[/FONT][FONT=MathJax_Main]±[/FONT][FONT=MathJax_Main]0.653271187094403 [/FONT][/FONT][FONT=MathJax_Main]cot equation solution[/FONT][FONT=MathJax_Main]

[/FONT][FONT=MathJax_Main]x=0...[/FONT][FONT=MathJax_Main][FONT=MathJax_Main][FONT=MathJax_Main]tan equation solution; j =[/FONT][/FONT][/FONT][FONT=MathJax_Main][FONT=MathJax_Main][FONT=MathJax_Main]0[/FONT][/FONT][/FONT][FONT=MathJax_Main]

and i dont know how to solv it ,, i get this from my example and i need to solve it for many others n examples so please help me.[/FONT]

First notice that if
\(\displaystyle \eta\, tan(h)\, +\, h\, =\, 0\)
then
\(\displaystyle -[\, \eta\, tan(h)\, +\, h\, ]\, =\, 0\)
or
\(\displaystyle \eta\, tan(-h)\, +\, (-h)\, =\, 0\)
So if +h is a solution, so is -h. Thus we only need speak of the positive solutions. Also, the solutions with the cotan in the equation would be handled about the same way as those for tan, so I will only discuss the equation with the tangent.

Next, it is possibly unfortunate I used n for the interval number since you may have confused it with the eta (\(\displaystyle \eta\)) of the problem so let me use a different symbol j. So, for \(\displaystyle \eta\ne0\), there is a single solution to the equation
\(\displaystyle \eta\, tan(h)\, +\, h\, =\, 0\)
in the interval \(\displaystyle (-\frac{\pi}{2}+ j\, \pi,\, \frac{\pi}{2}+ j\, \pi)\), j= 0, 1, 2, 3, ...
See the comments in red above

 

[dji321]

First notice that if
\(\displaystyle \eta\, tan(h)\, +\, h\, =\, 0\)
then
\(\displaystyle -[\, \eta\, tan(h)\, +\, h\, ]\, =\, 0\)
or
\(\displaystyle \eta\, tan(-h)\, +\, (-h)\, =\, 0\)
So if +h is a solution, so is -h. Thus we only need speak of the positive solutions. Also, the solutions with the cotan in the equation would be handled about the same way as those for tan, so I will only discuss the equation with the tangent.

Next, it is possibly unfortunate I used n for the interval number since you may have confused it with the eta (\(\displaystyle \eta\)) of the problem so let me use a different symbol j. So, for \(\displaystyle \eta\ne0\), there is a single solution to the equation
\(\displaystyle \eta\, tan(h)\, +\, h\, =\, 0\)
in the interval \(\displaystyle (-\frac{\pi}{2}+ j\, \pi,\, \frac{\pi}{2}+ j\, \pi)\), j= 0, 1, 2, 3, ...
See the comments in red above

ok is see now thnx ,, u tell me now that i can just resolve \(\displaystyle \eta\, tan(h)\, +\, h\, =\, 0\) ,,, if this ok i'm done with it ... plz tell me if i'm wrong ....

and maybe you can show me an example with lets take eta = 1 and we see the solutions ,, thank you again