Solve the initial value problems in Problems and graph each solution function x(t): x"+4x=delta(t)+delta(t-pi) x(0)=x'(0)=0

babeeb0oL 2020-12-17 Answered
Solve the initial value problems in Problems and graph each solution function x(t):
x"+4x=δ(t)+δ(tπ)
x(0)=x(0)=0
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Expert Answer

delilnaT
Answered 2020-12-18 Author has 94 answers
Step 1
Take a Laplace transform of the equation:
x"+4x=δ(t)+δ(tπ)
L(x(t))=X(s)
L(x")+L(4x)=L{δ(t)+δ(tπ)}   [L{δ(tc)}=ecs]
s2X(s)sx(0)x(0)+4X(s)=1+eπs
(s2+4)X(s)=1+eπs
X(s)=(1+eπs)s2+4
X(s)=1s2+4+eπss2+4
Step 2
Apply the inverse Laplace transform:
L1{X(s)}=L1{1s2+4+eπss2+4}
=L1{1s2+4}+L1{eπss2+4}
=12{L1{2s2+22}+L1{eπs2s2+22}}
=12L1{2s2+22}+12L1{eπs2s2+22}
Using: L1{ecsF(s)=u(tc)f(tc)}
=12sin(2t)+12u(tπ)sin(2(tπ))
=12sin(2t)+12u(tπ)sin(2t)
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