MATLAB: The function ‘solve’ doesn’t give all the solutions.

all solutionsfunctionsolve

% this does not give all the solutions
syms u;
see the short 3-line code there, try it out.
The solutions should be u1=-0.996308, and u2=6.63835
But this code only gives me u1, and mentions nothing about u2.
Can any one tells me why or which function should I use to get all the answers.

Best Answer

seems like you are interested in real solutions only. So use:
>> solve(1-0.99-(u+1)*exp(-u),'Real',true)
ans =
- lambertw(0, -exp(-1)/100) - 1
- lambertw(-1, -exp(-1)/100) - 1
If you dont set Real to true MATLAB solves it Complex which would give an infinite amount of solutions. When you use MuPAD Notepad you will obtain:
So only k=0 and k =-1 give Real solutions.
UPDATE: Since you want real solutions only its might be better to use a numeric solver:
>> feval(symengine,'numeric::solve','1-0.99-(u+1)*exp(-u)','AllRealRoots')
ans =
[ [u == -0.99630759688740599414645321772374], [u == 6.6383520679938122693712394050244]]