The given initial-value problem. y'' - 2y' - 8y = 5 y(0) = 1 y' (0) = 0

Yasmin 2020-11-03 Answered
The given initial-value problem.
y  2y  8y=5
y(0)=1
y(0)=0
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Expert Answer

Margot Mill
Answered 2020-11-04 Author has 106 answers
To solve, we use the properties of the Laplace transform and the Laplace table
The Laplace transform satisfies the linearity properties
L[f=g]=L[f] + L[g]
L[cf]=cL[f]
For all transfotmable functions f and g and constants c.
Laplace transform:
L[y  2y  8y]=L[5]
L[y]  2L[y]  8L[y]=5L[1]
Undefined control sequence \8
Y(s)(s2  2s  8)  y(0) + y(0)(2  s)=5s
Substitute initial conditions,
y(0)=1
y(0)=0
Solve for Y(s)
Y(s)(s2  2s  8)  0 + 1(2  s)=5s
Y(s)(s2  2s  8)  s + 2=5s
Y(s)(s2  2s  8)=5s  (s + 2)
=5  (s + 2)ss
Y(s)=s2  2s + 5s(s2  2s  8)
=58s + 1312(s + 2) + 1324(s  4)
Inverse transform:
L1[Y(s)]=L1[58s + 1312(s + 2) + 1324(s  4)]
= 58L1[1s] + 1312L1[1s  (2)] + 1324L[1s  4]
y(t)= 58 + 1312e2t + 1324e4t
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