Solve the initial value problem below using the method of Laplace transforms. y"-16y=32t-8e^{-4t} y(0)=0 y'(0)=15

ankarskogC 2021-02-21 Answered
Solve the initial value problem below using the method of Laplace transforms.
y"16y=32t8e4t
y(0)=0
y(0)=15
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Expert Answer

SoosteethicU
Answered 2021-02-22 Author has 102 answers
Step 1
Given, y"16y=32t8e4t
y(0)=0
y(0)=15
Step 2
y"16y=32t8e4t
Take Laplace transform of both sides of the equation
L{y16y}=L{32t8e4t}
L{y16y}:s2L{y}sy(0)y(0)16L{y}
L{32t8e4t}:32s28(s+4)
s2L{y}sy(0)y(0)16L{y}=32s28(s+4)
Plug in the initial conditions: y(0)=0,y(0)=15
s2L{y}s01516L{y}=32s28(s+4)
Simplify
s2L{y}16L{y}15=32s28(s+4)
Step 3
Isolate L{y}:
L{y}={15s3+52s2+32s+128s2(4+s)(s216)}
Take the inverse Laplace transform
y=L1{15s3+52s2+32s+128s2(4+s)(s216)}
y=2t2e4t+e4tt+2e4t
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