Solve the initial value problem below using the method of Laplace transforms y"-35y=144t-36^{-6t} y(0)=0 y'(0)=47

Zoe Oneal 2020-11-17 Answered
Solve the initial value problem below using the method of Laplace transforms
y"35y=144t366t
y(0)=0
y(0)=47
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Expert Answer

Raheem Donnelly
Answered 2020-11-18 Author has 75 answers
Solution:
The differential equation is
y"35y=144t366t
y(0)=0
y(0)=47
Apply Laplace transforms:
L{y"35y}=L{144t366t}
s2Y(s)sy(0)y(0)36Y(s)=144s236s+6
(s236)Y(s)47=144s236s+6
Y(s)=144s2(s236)36(s+6)(s236)+47(s236)
Decompose each term:
144s2(s236)=As+Bs2+Cs+6+Ds6
=As(s236)s2(s236)+B(s236)s2(s236)+Cs2(s6)s2(s236)+Ds2(s+6)s2(s236)
=(A+C+D)s3+(B6C+6D)s236As36Bs2(s236)
A=0,B=4,C=13,and D=13
Step 2
144s2(s236)=4s213(s+6)+13(s6)
36(s+6)(s236)=As+6+B(s+6)2+C(s6)
=A(s236)+B(s6)+C(s+6)2(s+6)(s236)
A=14,B=3,C=14
36(s+6)(s236)=14(s+6)+3(s+6)214(s6)
47(s236)=As+6+Bs6
A=4712, B=4712
Hence, 47(s236)=4712(s+6)+4712(s6)
Conclusion:
Hence, we have

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