Why did my analysis of critically damped motion lead to the wrong solution?

[LCSphysicist]

Homework Statement

ALl below

Relevant Equations

All below

Why is my analysis of critically damped motion wrong?
x'' + y*x' + wo²x = 0

Choosing a complex number z as z = A*e^i(wt+a) and putting on the equation calling x as the real part of Z

w = ( i*y +- (4wo²-y²)^(1/2) )/2 (bhaskara)

2wo = y (critical)

w = iy/2

z = A*e^i(ity/2 + a)
z = A*e^(-yt/2 + a)
x = A*e^(-yt/2)*cos(a)

 

[LCSphysicist]

In the rolling of this question, i think the wrong step i did was adopt z as i adopted, but, if is this, raised up another question:
I am still a beginner dealing with this type of solution of differential equation by complex, so, how would i know the right z i need to assume before starting the mathematical steps?

 

[ehild]

In the rolling of this question, i think the wrong step i did was adopt z as i adopted, but, if is this, raised up another question:
I am still a beginner dealing with this type of solution of differential equation by complex, so, how would i know the right z i need to assume before starting the mathematical steps?

In case of critically damped or overdamped oscillators, the solution is not complex.
Read about the solutions of linear, constant- coefficient homogeneous ODE-s. (https://www.math24.net/second-order-linear-homogeneous-differential-equations-constant-coefficients/ , for example)
Usually, we start solving such equations by assuming the solution in exponential form $y=e^{\lambda t}$, substituting back into the ODE and getting the characteristic equation for lambda, a quadratic equation, which has either complex roots (underdamped oscillator) or two different real roots (overdamped oscillator) or a double root in case of critical damping.
While the general solution is linear combination of the different exponentials, the solution is $e^{\lambda t}(c_1+c_2t)$ in case of double root, that is critical damping.