Two different solutions for one differential equation (population model)

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I'll try solving the ODE:
Substituting
Transforming to:
Solving I get:
Finally, after back substitution:
complete solution:
what's equivalent to:
Now same stuff with MATLAB:
syms u(t); syms c1 c2 u0 real;
D = diff(u,t,1) == c1*u-c2*u^2;
k2 = u;
cond = k2(0) == u0;
S = dsolve(D,cond);
pretty(S)
Receiving:
I was hoping these expressions have some equivalence so I was plotting them:
c1 = 4; c2 = 2; u0 = 1;
syms t
P1 = (c1)/(1-exp(-c1*t)+c1/u0*exp(-c1*t));
fplot(P1)
hold on
P2 = -(c1*(tanh(atanh((c1 - 2*c2*u0)/k1) - (c1*t)/2) - 1))/(2*c2);
fplot(P2)
but no luck there. I know that's again a quite complex question, but on MathStack one told me these solutions are equvialent, so I don't see a reason for the dissonance.
  3 Comments
Sulaymon Eshkabilov
Sulaymon Eshkabilov on 3 Jun 2021
Most welcome. We learn by making mistakes.
Please just keep it. So others can learn.

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Accepted Answer

Sulaymon Eshkabilov
Sulaymon Eshkabilov on 29 May 2021
Besides k1, in your derivations, there are some errs. Here are the corrected formulation in your derivation part:
c1 = 4; c2 = 2; u0 = 1;
syms t
P1 = c1/(c2 - exp(-c1*t)*(c2 - c1/u0)); % Corrected one!
fplot(P1, [0, pi], 'go-')
hold on
P2 = -(c1*(tanh(atanh((c1 - 2*c2*u0)/c1) - (c1*t)/2) - 1))/(2*c2);
S = eval(S);
fplot(S, [0, pi], 'r-')
Good luck.

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