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Solving the SHM differential equation
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[QUOTE="kuruman, post: 6829393, member: 192687"] All of the expressions below are general solutions of your equation [LIST=1] [*]##x=C_1e^{i\omega t}+C_2e^{-i\omega t}## [*]##x=A\sin\omega t+B\cos\omega t## [*]##x=D\sin(\omega t+\phi)## [/LIST] You can verify that this is so by substituting in your ODE. Note that each expression has two arbitrary constants that are determined by the initial conditions, usually the values of ##x## and ##\frac{dx}{dt}## at ##t=0## that are appropriate to a particular situation.. You are asking how to go from 5 to 6 which is essentially going from my item 2 to 3. It is more obvious to see how to go from 3 to 2. Once you see that, you can reverse the algebra, if you wish. Using a well known trig identity for the sine of a sum of angles, $$D\sin(\omega t+\phi)=D\cos\phi \sin\omega t+D\sin\phi \cos\omega t.$$ If you identify $$A\equiv D\cos\phi~~\text{and}~~B\equiv D\sin\phi,$$you have item 2 above. [/QUOTE]
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Solving the SHM differential equation
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