# Solve can't find a solution for a transcendental equation

I'm trying to solve a system of equations. The problem I'm encountering is that one equation is transcendental and Solve is unable to find a solution for it, so can't find a solution for the system. Is there any way to solve this system?

Smallest Working Example:

BearLM = {{1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 10^7, 1, 0, 0}, {-500,
0, 0, 1, 0}, {0, 0, 0, 0, 1}};
RollLM[q_] = {{1, 20, Power[20, 2]/(2* 4.2*10^13), Power[20, 3]/(
6 *4.2*10^13), q Power[20, 4]/(24* 4.2*10^13)}, {0, 1, 20/(
4.2*10^13), Power[20, 2]/(2* 4.2*10^13),
q Power[20, 3]/(6 *4.2*10^13)}, {0, 0, 1, 20,
q Power[20, 2]/2}, {0, 0, 0, 1, q 20}, {0, 0, 0, 0, 1}} ;
Vec11 = {vA, \[CurlyPhi]A, 0, 0, 1};
Vec12 = BearLM. Vec11;
VecList1 = {Vec12};
Vec21 = {vS, \[CurlyPhi]S, 0, 0, 1};
Vec22 = BearLM.Vec21;
VecList2 = {Vec22};
Gls := {
{vAR, \[CurlyPhi]AR, FQAR, MbAR, 1} ==
RollLM[qW + 1].Last[VecList1],
{vSR, \[CurlyPhi]SR, FQSR, MbSR, 1} == RollLM[-qW].Last[VecList2],
vAR - vSR == 8.5*10^(-6) qW*Log[12000/(4.6*10^(-5) qW)]
};

Res = Solve[
Gls, {vAR, \[CurlyPhi]AR, FQAR, MbAR, vSR, \[CurlyPhi]SR, FQSR,
MbSR, qW}][[1]]

• Feb 22, 2019 at 19:26

eqs = Flatten[Thread /@ Gls]
vars = {vAR, \[CurlyPhi]AR, FQAR, MbAR, vSR, \[CurlyPhi]SR, FQSR, MbSR, qW}


Solve all but the last equations (linear system) in terms of all vars but qW:

partial = Solve[eqs[[1 ;; -2]], vars[[1 ;; -2]]];


The remaining equation is:

last = Simplify[eqs[[-1, 1]] - eqs[[-1, 2]] /. First[partial]]


It can apparently be handled by Solve now :

qwsol = Solve[last == 0, vars[[-1]]]


Quick check:

FullSimplify[last /. First[qwsol]] /. {vA -> 1.1, vS -> 2.2, \[CurlyPhi]A -> 3.3, \[CurlyPhi]S -> -4.4 }
(* -5.9992*10^-12 *)