Use Laplace transforms to solve the following initial value problem y"-y'-6y=0 y(0)=1 y'(0)=-1

facas9 2021-02-14 Answered
Use Laplace transforms to solve the following initial value problem
y"y6y=0
y(0)=1
y(0)=1
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Expert Answer

Maciej Morrow
Answered 2021-02-15 Author has 98 answers
Step 1
The given IVP is as follows.
y"y6y=0
y(0)=1
y(0)=1
Apply Laplace transform on the IVP as follows.
yy6y=0
L{y}L{y}L{6y}=L{0}
s2L{y}sy(0)y(0)sL{y}+y(0)6L{y}=0
L{y}[s2s6]=s(1)+(1)(1)
L{y}[s2s6]=s2
L{y}=s2s2s6
L{y}=s2(s3)(s+2)
Step 2
Apply partial fraction on s2(s3)(s+2) as follows.
s2(s3)(s+2)=As3+Bs+2
s2=A(s+2)+B(s3)
s2=(A+B)s+(2A3B)
A+B=1,2A3B=2
A=1B
2(1B)3B=2
5B=4
B=45
A=15
s2(s3)(s+2)=15(1s3)+45(1s+2)
Obtain the solution of the IVP as follows.
L{y}=15(1s3)+45(1s+2)
Take inverse Laplace transforms as follows.
L1{L{y}}=L1{15(1s3)+45(1s+2}

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