Solve the given initial-value problem.d2xdt2=ω2x=F0cos⁡(γt), x(0)=0, x′(0)=0

ajedrezlaproa6j

ajedrezlaproa6j

Answered

2021-12-30

Solve the given initial-value problem.
d2xdt2=ω2x=F0cos(γt), x(0)=0, x(0)=0

Answer & Explanation

Timothy Wolff

Timothy Wolff

Expert

2021-12-31Added 26 answers

Find the homogeneous solution of the given IVP as follows.
The given IVP is as follows,
d2xdt2=ω2x=F0cos(γt), x(0)=0, x(0)=0 and ωγ
x+ω2x=F0cos(γt), x(0)=x(0)=0 and ωγ
Consider the homogeneous part x+ω2x=0
The characterice equation is m2+ω2=0
m2+ω2=0
⇒=ω
⇒=ωi,ωi
The homogeneous solution is xh=C1cos(ωt)+C2sin(ωt)
Find the particular solution of the given IVP as follows.
The IVP is x+ω2x=F0cos(γt), x(0)=x(0)=0 and ωγ
Let the particular solution be of the form, xp=Acos(γt)
x=γAsin(γt)
x=γ2cos(γt)
γ2cos(γt)+ω2Acos(γt)=F0cos(γt)
Aω2γ2=F0
A=F0+γ2ω2,ω0
The particular solution is xp=F0+γ2ω2cos(γt)
The general solution is of the given IVP is as follows.
The general solution is
x(t)=xh+xp
=C1cos(ωt)+C2sin(ωt)+F0+γ2ω2cos(γt)

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