Solve the following initial value problem by using Laplace transform. y"+4y=4t+12 , y(0)=4 , y'(0)=0

pancha3 2021-09-03 Answered

Solve the following initial value problem by using Laplace transform
y+4y=4t+12,y(0)=4,y(0)=0

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Expert Answer

odgovoreh
Answered 2021-09-04 Author has 107 answers

y+4y=4t+12,y(0)=4,y(0)=0
Taking the Laplace transform on both side
L{y"}+4L{y}=4L{t}+12L{1}
s2L{y}sy(0)y(0)+4L{y}=41s2+121s
(s2+4)L{y}4s0=4(3s+1s)
(s2+4)L{y}=4(3s+1s+s)
(s2+4)L{y}=4(3s+1+s2s)
L{y}=4(s2+3s+1s(s2+4))
=4(s2+4+3s3s(s2+4))
=4[1s+3s3s(s2+4)]
=4[1s+14(12s12s(s2+4))]
=4[1s+14(3s23s2+12s12s(s2+4))]
=41s+s(3s+12)3(s2+4)s(s2+4)
L{y}=4s+3s+12s2+43s
L{y}=1s+3ss2+4+12s2+4
y(t)=L1{1s}+3L1{ss2+4}+6L1{2s2+4}
y(t)=1+3cos(2t)+6sin(2t)

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