Simple Harmonic Motion and the Corresponding 2nd Order Differential Eq.

[sam1996]

Hi everybody,

I'm doing a piece of work on second order differential equations and their use in simple harmonic motion. Using this as resource, http://tutorial.math.lamar.edu/Classes/DE/Vibrations.aspx , I do not understand how the author reaches equation (4). Does it include the use of Euler's formula? I've tried working it on paper, but haven't managed anything useful...

Here's what I did so far anyway:

I assumed a solution in the form of e^rt, managed to get the correct characteristic equation and the correct complex roots to said characteristic equation, as shown in the resource. Then, I could not use these complex conjugate roots to get the answer shown in the resource. Would you start by writing +/- e^(iwt), with w as omega, as cos(wt)+isin(wt) and cos(wt)-isin(wt). Correct so far? What would you then do to find the most general solution?

Many thanks for your help!

Sam

 

[Ishuda]

Hi everybody,

I'm doing a piece of work on second order differential equations and their use in simple harmonic motion. Using this as resource, http://tutorial.math.lamar.edu/Classes/DE/Vibrations.aspx , I do not understand how the author reaches equation (4). Does it include the use of Euler's formula? I've tried working it on paper, but haven't managed anything useful...

Here's what I did so far anyway:

I assumed a solution in the form of e^rt, managed to get the correct characteristic equation and the correct complex roots to said characteristic equation, as shown in the resource. Then, I could not use these complex conjugate roots to get the answer shown in the resource. Would you start by writing +/- e^(iwt), with w as omega, as cos(wt)+isin(wt) and cos(wt)-isin(wt). Correct so far? What would you then do to find the most general solution?

Many thanks for your help!

Sam

Your general solution is
f(x) = A e^iwx + B e^-iwx
So, use Euler's formula
f(x) = A [cos(wx) + i sin(wx)] + B [cos(wx) - i sin(wx)]
or
f(x) = (A + B) cos(wx) + i (A-B) sin(wx)
Now, let
C = A+B
D = i (A-B)
to get a general solution of
f(x) = C cos(wx) + D sin(wx)
Since A and B were arbitrary constants, we make take C and D as arbitrary constants.

 

[sam1996]

Many thanks Ishuda, makes perfect sense!! Fantastic help!!!

Your general solution is
f(x) = A e^iwx + B e^-iwx
So, use Euler's formula
f(x) = A [cos(wx) + i sin(wx)] + B [cos(wx) - i sin(wx)]
or
f(x) = (A + B) cos(wx) + i (A-B) sin(wx)
Now, let
C = A+B
D = i (A-B)
to get a general solution of
f(x) = C cos(wx) + D sin(wx)
Since A and B were arbitrary constants, we make take C and D as arbitrary constants.