Tuesday, 8 March 2016

homework and exercises - Harmonic oscillator differential equation solution




My math book explains how to solve second order equations like :
¨x+ω2x=0


but I end up with the general solution : Acos(ωt)+iBsin(ωt).


Now my physics book says the solution is ρcos(ωt+ϕ)


How can I get there from the general solution?



Answer



Using the sum rule for cosines, we find ρcos(ωt+ϕ)=ρcos(ϕ)cos(ωt)ρsin(ϕ)sin(ωt). So we see that ρcos(ωt+ϕ) is the same as Acos(ωt)+iBsin(ωt) when A=ρcos(ϕ) and B=iρsin(ϕ).


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