Use the Laplace transform to solve the given initial-value problem. displaystyle{y}{''}-{6}{y}'+{13}{y}={0}, {y}{left({0}right)}={0}, {y}'{left({0}right)}=-{9}

Globokim8 2020-10-28 Answered
Use the Laplace transform to solve the given initial-value problem.
y6y+13y=0,   y(0)=0,   y(0)=9
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Expert Answer

Jayden-James Duffy
Answered 2020-10-29 Author has 91 answers

Step 1
The given differential equation is
y6y+13y=0
Step 2
Simplify the initial value problem using Laplace transform:
L(y6y+13y)=L(0)
L(y)6L(y)+13L(y)=0
s2Y(s)sy(0)y(0)6(sY(s)y(0))+13Y(s)=0
s2Y(s)+96sY(s)+13Y(s)=0
Y(s)=9s26s+9+4
Step 3
Simplify further,
Y(s)=9(s3)2+4
Y(s)=9(s3)2+22
Take inverse laplace on both sides,
L1(Y(s))=L1(9(s3)2+22)
y(t)=92L1(2(s3)2+22)
y(t)=92e3tsin2t
Hence , the solution of the initial value problem is y(t)=92e3tsin2t

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