Warning: Cannot find explicit solution

Hi, I have to solve this equation: x*cot(x)-1+(ke*R/Dm). Where ke= 3.7*10^-5, R=2*10^-4 and Dm=12.6*10^-19. When I use solve matlab give me this error: 'Warning: Cannot find explicit solution.'. Can you help me?

Answer 1

[John D'Errico]

solve is telling you that no analytical solution exists. That is what you tried to do, is it not? Do you assume that all equations have such a solution?

First, I would recommend that you learn to use scientific notation in the form I show. It will make your code easier to read, write, and follow.

ke = 3.7e-5;
R = 2e-4;
Dm = 12.6e-19;
ratio = ke*R/Dm

ratio =
  5.873e+09

So, first, note that the ratio you have written is quite a large number in comparison to 1, yet you are then subtracting 1 from that. That itself will cause a potential problem in terms of accuracy, since MATLAB will compute that result as a DOUBLE precision number, then subtract 1.

In fact, see that you get a different result even if you do these seemingly similar things:

ratio = sym(ke*R/Dm)
ratio =
1539575873015873/262144

ratio = sym(ke)*R/Dm
ratio =
24014372970723576557818871808/4088933776096176875

syms x
fun = x*cot(x)-1 + ratio;

As you should expect, this will have infinitely many zeros, since it contains the cot function.

ezplot(fun)

The first real positive solution seems to lie roughly near pi from the plot, but vpasolve does not know which of infinitely many solutions it should search for.

vpasolve(fun)
ans =
-6029099.5755159751256227152413773

Telling it to look near pi helps.

vpasolve(fun,[2 4])
ans =
3.1415926530548734082568136811652

format long g
pi
ans =
          3.14159265358979

So just a tiny bit less than pi, if you know the first dozen digits of pi.

[Valentina Comparelli]

thank you so much :D

Answer 2

[Torsten]

format long
deltax = 1e-13;
n = 100;
ke = 3.7e-5;
R = 2e-4;
Dm = 12.6e-19;
fun = @(x)x.*cot(x)-1+ke*R/Dm;
for i=1:n
  left=(i-1)*pi+deltax;
  right=i*pi-deltax;
  sol(i)=fzero(fun,[left right]);
end
sol
fun(sol)