# Alternative way for getting the roots of a transcendental equation and verifying them

I have the following transcendental equation:

$$2 \cot(x)=\frac{kx}{h(L/N)}-\frac{h(L/N)}{kx}$$

I use the following code to solve it

roots =
Sort[x /.
NSolve[
{2*Cot[x] == Rationalize[k/(h (L/N))] x - Rationalize[h (L/N)/k]/x, 60 > x > 0},
x, Reals]]


Typical values for the constants are $$L=0.25,N=20,k=16,h=0.1$$:

Is there an alternative way to solve this equation and check whether the answers from both the methods match?

FindInstance[2*Cot[x] == (k x)/(h (L/Nd)) - h (L/Nd)/(k x), {x}, Reals, 10]


{{x -> -0.0124999}, {x -> -0.0124999}, {x -> 0.0124999}, {x -> 0.0124999},
{x -> 0.0124999}, {x -> -0.0124999}, {x -> -0.0124999}, {x -> 0.0124999},
{x -> -0.0124999}, {x -> 0.0124999}}

• FindInstance – LouisB Dec 27 '19 at 8:20
• @LouisB I tried FindInstance but it manages to find only one root. This root matches with theNSolve result. Is there a workaround to this ? Like if I only want positive roots. Giving the argument to FindInstance for 10 roots leads to repeated roots. I have added my attempt to the original question. – Indrasis Mitra Dec 27 '19 at 9:56
• Try it this way FindInstance[ { 2*Cot[x] == k/(h (L/Nd)) x - h (L/Nd)/k/x, 60 > x > 0 }, x, Reals, 30] – LouisB Dec 27 '19 at 10:12
• @LouisB Thanks. It works perfectly. If you can add this as an answer, I am willing to accept it. It solves my problem. – Indrasis Mitra Dec 27 '19 at 10:15

Clear["Global*"]

roots[Lv_?NumericQ, nv_?NumericQ, kv_?NumericQ, hv_?NumericQ] :=
Module[{L, n, k, h},
{L, n, k, h} = Rationalize[{Lv, nv, kv, hv}, 0]; Solve[{
2*Cot[x] == k/(h (L/n)) x - h (L/n)/k/x,
60 > x > 0}, x, Reals]]


The exact solutions are Root objects

sol1 = roots[0.25, 20, 16, 0.1] These are approximately

sol1 // N

(* {{x -> 0.0124999}, {x -> 3.14164}, {x -> 6.28321}, {x -> 9.42479}, {x ->
12.5664}, {x -> 15.708}, {x -> 18.8496}, {x -> 21.9912}, {x ->
25.1327}, {x -> 28.2743}, {x -> 31.4159}, {x -> 34.5575}, {x ->
37.6991}, {x -> 40.8407}, {x -> 43.9823}, {x -> 47.1239}, {x ->
50.2655}, {x -> 53.4071}, {x -> 56.5487}, {x -> 59.6903}} *)


You can verify the solutions by substituting back into the equations

2*Cot[x] == k/(h (L/n)) x - h (L/n)/k/x &&
60 > x > 0 /.
Rationalize[{0.25, 20, 16, 0.1}]] /.
sol1 // FullSimplify

(* {True, True, True, True, True, True, True, True, True, True, True, True, \
True, True, True, True, True, True, True, True} *)


Using FindInstance as the second method

roots2[Lv_?NumericQ, nv_?NumericQ, kv_?NumericQ, hv_?NumericQ,
nmbr_Integer?Positive] := Module[{L, n, k, h},
{L, n, k, h} = Rationalize[{Lv, nv, kv, hv}, 0];
FindInstance[
2*Cot[x] == k/(h (L/n)) x - h (L/n)/k/x &&
60 > x > 0, x, Reals, nmbr]]

sol2 = roots2[0.25, 20, 16, 0.1, Length[sol1]];


Verifying that the exact solutions are identical

sol1 === sol2

(* True *)

• Thanks. This was very descriptive. – Indrasis Mitra Dec 28 '19 at 4:40

Your equation might be transformed to 2 Cot[x] == x/p - p/x with a new parameter p= (h (L/n) )/k

With

p0=(h (L/n) )/k /. {L -> 0.25, n -> 20, k -> 16, h -> 0.1};
(*0.000078125*)


you might visualize the solution with ContourPlot

ContourPlot[ContourPlot[1/(2 Cot[x]) == 1/((x/p) - p/x), {x, 0, 60}, {p, 0, 2 p0},FrameLabel -> {x,"p=\!$$\*FractionBox[\(\(h$$$$\\\$$$$(L/n)$$$$\\\$$\), \$$k$$]\)"}], {x, 0, .01 Pi /2}, {p, 0, 2 p0},
FrameLabel -> {x,"p=\!$$\*FractionBox[\(\(h$$$$\\\$$$$(L/n)$$$$\\\$$\), \$$k$$]\)"}] The solution for given p=p0follow to

NSolve[{2 Cot[x] == x/p0 - p0/x, 0 < x < 60}, x, Reals]
(*{{x -> 0.0124999}, {x -> 3.14164}, {x -> 6.28321}, {x ->9.42479}, {x -> 12.5664}, {x -> 15.708}, {x -> 18.8496}, {x ->21.9912}, {x -> 25.1327}, {x -> 28.2743}, {x -> 31.4159}, {x ->34.5575}, {x -> 37.6991}, {x -> 40.8407}, {x -> 43.9823}, {x ->47.1239}, {x -> 50.2655}, {x -> 53.4071}, {x -> 56.5487}, {x ->59.6903}}*)

• Thanks for the visualizations. I was already using NSolve` to find the roots and was looking for an alternative. – Indrasis Mitra Dec 28 '19 at 3:54