In Physics, the Simple Harmonic Oscillator is represented by the equation d^{2}x/dt^{2}=-\omage^{

Pam Stokes 2022-01-16 Answered

In Physics, the Simple Harmonic Oscillator is represented by the equation d2x/dt2=ω2x.
By using the characteristic polynomial, you get solutions of the form x(t)=Aeiωt+Beiωt, I get that you Euler's formula eiθ=cosθ+isinθ, but I can't seem to find my way all the way to the ''traditional form'' of Dcosωt+Csinωt.

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kalupunangh
Answered 2022-01-17 Author has 29 answers

However, if you assume the function x(t) is real, then they are related as A=B.
This can be seen since x(t)=x(t) which gives us A=B (since eiωt and eiωt are orthonormal)
Rewriting the complex exponentials as sine and cosine we get
x(t)=A(cos(ωt)+isin(ωt))+B(cos(ωt)isin(ωt))=(A+B)cos(ωt)+i(AB)sin(ωt)
Now as proved before A=B and hence (A+B) is real and (AB) is imaginary and hence we can write (A+B)=C and i(AB)=D where C, D are real number.
And the solution becomes
x(t)=Ccos(ωt)+Dsin(ωt)
where C,DR

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Robert Pina
Answered 2022-01-18 Author has 42 answers

Remember that x(t)=Aeiωt+Beiωt+Beiωt is a solution for any values of A and B which means that choosing A=B=12 result in the solution
x1(t)=eiωt+eiωt
Euler's fomula, along with the properties cos(ωt)=cos(ωt) and sin(ωt)=sin(ωt), simplifies this solution to
x1=cos(ωt)
Similarly, choosing A=1/(2i) and B=1/(2i) gives the solution x2(t)=sin(ωt)
Finally, by linearity you can conclude that the linear combination Cx1+Dx2 is also a solution.

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alenahelenash
Answered 2022-01-24 Author has 368 answers
the answers given are absolutely fine as they show the way from e±iωt to sin and cos. It might be even shorter if you use that sin(x)=12i(eixeix) and cos(x)=12(eix+eix) This can be interpreted as a definition for the functions or seen from its series definitions.
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