Ansatz for Hysteretic oscillation

[Remixex]

Homework Statement

Given
X'' + iG^2 X + w^2 X = 0
G^2 = h/m
w^2= k/m
What would be a good "educated guess" to solve that differential equation?
My oscillations and waves teacher asked this on a test and since I didn't see anything depending on the speed of the object X assumed it was a really weird kind of Simple Harmonic Motion.
We were all wrong x.x
I actually don't know if to ask this here or in the math forums

Homework Equations

The Attempt at a Solution

Ansatz= Acos(ϑt + ϒ)[/B][/B]
It was wrong, i tried to factor iG^2 and w^2 into one big "harmonic" frequency and solve like an SHM

 

[rude man]

You need to explain what the system is you're describing with the equation.
What is X? displacement?
what is G? what does h/m mean? I know m is mass but what is h?
is i = √(-1)?
Is "Ansatz" your guess as to the solution? not a commonly used term, at least not in the U.S.

 

[Remixex]

Ansatz means educated guess, you "guess" the solution to the differential equation

a more general form would be
mX'' + hXi + kX = 0 with m mass, h hysteretic coefficient and k spring constant (all are real constants)
I just don't know how to solve a differential equation with a imaginary (i) number within it, every time i search it takes me into Real damped systems with imaginary roots (but not imaginary coefficients)
i tried solving the system like a harmonic oscillation and it was wrong

 

[rude man]

You may need to take a course in solving differential equations. I don't see that the fact that one of the coefficients of a linear ordinary differential equation (ODE) with constant coefficients is imaginary should be a problem. There is formal math for handling complex numbers.

Anyway, your Ansatz is wrong; it applies to a system without damping, i.e. h = 0.

Since you may not have a background in solving ODE's, look up the solution to the second-order system with damping, i.e. x'' + bx' + kx = 0. You also need a finite initial condition if you want the non-trivial solution (trivial is x = 0).

You also haven't described your system verbally which sort of ties our hands. It's hard for me to imagine that making the damping term imaginary suffices to describe a system with hysteresis. A system with hysteresis requires very advanced math, called "describing functions".