Forest UCM Osc SHM

Equation of motion

In solving the differential equation

[math]\displaystyle{ m\ddot x =-kx }[/math]energy is constant with time

and observe that the above differential equation is a special case of the more general differential equation

[math]\displaystyle{ m\ddot x + c \dot x =-kx }[/math]energy is constant with time

one could rewrite the above as

[math]\displaystyle{ \left ( \frac{d}{dt}\frac{d}{dt} +\frac{d}{dt} + k \right ) x = 0 }[/math]

One could cast the above differential equation into an analogous quadratic equation if you

let

[math]\displaystyle{ O = \frac{d}{dt} }[/math]

then the analogous equation becomes

[math]\displaystyle{ \left ( mO^2 + aO + k \right ) x = 0 }[/math]

where m, a, and k are constants

factoring this quadratic you would have

[math]\displaystyle{ \left ( O - \gamma \right )\left ( O - \beta \right ) x = 0 }[/math]

where a non-trivial solution would exist if one of the terms in the parentheses were zero

this basically reduces our 2nd order differential equation down to two first order differential equations

[math]\displaystyle{ \left ( \frac{d}{dt} - \gamma \right ) \left ( \frac{d}{dt} + \beta \right ) x = 0 }[/math]

one of the solutions would be

[math]\displaystyle{ \frac{d}{dt} x - \gamma x= 0 }[/math]

[math]\displaystyle{ \frac{dx}{x} = \gamma dt }[/math]

[math]\displaystyle{ x = Ae^{\gamma t} }[/math]

[math]\displaystyle{ x = Ae^{\gamma t} + Be^{\beta t} }[/math]

For the special case where there isn't a first derivative term (a=0)

You simply have

[math]\displaystyle{ \left ( m\frac{d}{dt}\frac{d}{dt} + k \right ) x = 0 }[/math]

or

[math]\displaystyle{ \left ( mO^2 + k \right ) x = 0 }[/math]

[math]\displaystyle{ O= \pm \sqrt{-\frac{k}{m}} = \pm i \sqrt{-\frac{k}{m}} }[/math]

[math]\displaystyle{ \gamma \equiv +i\sqrt{\frac{k}{m}} = i \omega }[/math]

[math]\displaystyle{ \beta = -i\sqrt{\frac{k}{m}} = -i \omega }[/math]

then you have

[math]\displaystyle{ x = Ae^{\gamma t} + Be^{\beta t} }[/math]

[math]\displaystyle{ = Ae^{i \omega t} + Be^{-i\omega t} }[/math]

Forest_UCM_Osc#Simple_Harmonic_Motion_.28SHM.29