Equation of motion solutions

[davesface]

1. The problem statement, all variables and given/known data
Find the equation of motion for a particle of mass m subject to a force F(x)=-kx where k is a positive constant. Write down the equation of motion as x''(t)=F/m. Then show that x(t)=Ceiwt is a solution to the equation of motion for any C as long as w has one of 2 possible values (i is the imaginary unit, w is omega, t is time). What are those values?

There's more to it, but I am totally lost as to how I can at least start from this information.

2. Relevant equations
x''(t)=F/m
F(x)=-kx, where k is a positive constant
x(t)=Ceiwt

3. The attempt at a solution
I took the derivative of the last equation listed in b twice to get x'(t)=iwCeiwt and then x''(t)=i2w2Ceiwt, which simplifies to x''(t)=-w2Ceiwt.

I guess that I really would just like to know if I'm anywhere in the ballpark for how the problem should begin. It's not a graded problem, but I hate leaving it unsolved.

[HallsofIvy]

1. The problem statement, all variables and given/known data
Find the equation of motion for a particle of mass m subject to a force F(x)=-kx where k is a positive constant. Write down the equation of motion as x''(t)=F/m. Then show that x(t)=Ceiwt is a solution to the equation of motion for any C as long as w has one of 2 possible values (i is the imaginary unit, w is omega, t is time). What are those values?

There's more to it, but I am totally lost as to how I can at least start from this information.

2. Relevant equations
x''(t)=F/m
F(x)=-kx, where k is a positive constant
x(t)=Ceiwt

3. The attempt at a solution
I took the derivative of the last equation listed in b twice to get x'(t)=iwCeiwt and then x''(t)=i2w2Ceiwt, which simplifies to x''(t)=-w2Ceiwt.

I guess that I really would just like to know if I'm anywhere in the ballpark for how the problem should begin. It's not a graded problem, but I hate leaving it unsolved.
Looks to me like you are doing the problem backwards! You are first asked to write down the equation of motion. You give as "relevant equations" x"= F/m and F= -kx. Okay, looks to me like the equation of motion is x"= -kx/m.

NOW you can argue that if x= Ceiwt, then x'= Ciweiw and x"= -Cw2eiwt= -w2(Ceiwt which is the same as -kx/m as long as w2= -k/m. That last equation should tell you what values w can have.

[davesface]

x"= -Cw2eiwt= -w2(Ceiwt which is the same as -kx/m as long as w2= -k/m. That last equation should tell you what values w can have.

2 questions there:
1. Why is Ceiwt which is the same as -kx/m as long as w2= -k/m?
2. How does w2= -k/m lead me to the values of w?