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The peak response of an underdamped second order system is:

Mpt = 1 + e^((-z*pi)/sqrt(1-z^2))

Where Mpt is real.

For Mpt = 5, the command:
solve( 5 = 1 + e^((-z*pi)/sqrt(1-z^2)) ,z)

Returns:



Which is only the negative result. The same happens in the Solve app, I only get the negative result.

The same code in matlab:

clc; syms z;
eqn = 5 == 1+ exp((-z*pi)/sqrt(1-z^2));
vpa(solve(eqn,z))

produces:

ans =
0.40371275194342068038575605143555
-0.40371275194342068038575605143555

Try both answers from matlab in your expression. :-)

What is inside e^(...) must be positive, so (-z*pi)/sqrt(1-z^2) > 0, this imply z < 0

Matlab positive z is a mistake. (Probably just return solved quadratic roots)

Xcas 1, Matlab 0 :-)