How can I solve this?

clc

clear

syms PL c k

eqn = [PL * exp(-c*exp(-k*0)) == 179323 ; PL * exp(-c*exp(-k*10)) == 203302 ; PL * exp(-c*exp(-k*20)) == 226542]

S = solve(eqn , [PL;c;k] )

S =

struct with fields:

PL: [0×1 sym]

c: [0×1 sym]

k: [0×1 sym]

[Muhammad Usman]

the set of equations are non linear in nature that's why you can't use solve to compute the solution

[Walter Roberson]

The equations have no solution over reals.

Answer 1

[Muhammad Usman]

% Solve the system of equations starting at the point [0,0,0].
% PL = x(1); c = x(2); k = x(3);
% Initial guess is [0,0,0], you can change it accordingily
fun = @root2d;
x0 = [0,0,0];
x = fsolve(fun,x0)

Warning: Trust-region-dogleg algorithm of FSOLVE cannot handle non-square systems; using Levenberg-Marquardt algorithm instead.

No solution found.

fsolve stopped because the relative size of the current step is less than the
value of the step size tolerance, but the vector of function values
is not near zero as measured by the value of the function tolerance.

x = 1×3

1.0e+05 * 2.0192 -0.0000 0

function F = root2d(x)
F(1) = x(1) * exp(-x(2)*exp(-x(3)*0)) - 179323; 
F(2) = x(1) * exp(-x(2)*exp(-x(3)*10)) - 203302; 
F(2) = x(1) * exp(-x(2)*exp(-x(3)*20)) - 226542;
end

[CHENG WEI LI]

The function has correct answer.

why this answer is wrong

[Alex Sha]

There are actually numerical solutions like below:

x1: 446505.431672107
x2: 0.912262916225993
x3: 0.0148006249649759

Answer 2

[Torsten]

syms PL c k 
eqn1 = PL * exp(-c*exp(-k*0)) == 179323;
eqn2 = PL * exp(-c*exp(-k*10)) == 203302;
eqn3 = PL * exp(-c*exp(-k*20)) == 226542;
sPL = solve([eqn1 eqn2],[c,k]);

Warning: Solutions are only valid under certain conditions. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'.

PLsol = solve(subs(eqn3,[c,k],[sPL.c sPL.k]),PL);
sck = solve([subs(eqn1,PL,PLsol),subs(eqn2,PL,PLsol)],[c,k]);
csol = sck.c;
ksol = sck.k;
vpa(PLsol)

ans = 

446505.43167210849515476976968181

vpa(csol)

ans = 

0.91226291622599614437018856917074

vpa(ksol)

ans = 

0.014800624964975891465693800088887