solve implicit function




On Mon, 27 Apr 2009, lars b?ttcher wrote:

< Hi,
< I would like to solve an equation with maximas's solve-function. The problem 
< is that it is an implicit function and the solve-function give no result. How can I 
< solve this Problem. 
< 
< 
< (%i1)  eqn3: 1/w*sqrt(Rs(rho_c+0.2*Rs*h^2))*coth(sqrt(Rs/(rho_c+0.2*Rs*h^2))*d)-Rc;
< 
< (%i2) Rs: 2.53;
< 
< (%i3) Rc: 0.395;
< 
< (%i4) h: 1.4e-6;
< 
< (%i5) d: 49e-6;
< 
< (%i6) w: 100e-6;
< 
< (%i7) solve(eqn3,rho_c);
 
I think there is a typo in %i1. 

You want to solve something like
s*coth(d*s) = c

for s and then solve

s = sqrt(Rs/(rho_c+0.2*Rs*h^2))

for rho_c.

The first equation is transcendental (search for 'Lambert's W-function'),
so you'll need to be content to solve it numerically, since Maxima
doesn't know about its multi-valued solution. 

See Ted Woollett's book chapter 4, section 4.2.3, for an explanation of
how to do that.

http://www.csulb.edu/~woollett/mbe4solve.pdf 

Leo
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