Use the Laplace transform to solve the given initial-value problem {y}{''}+{2}{y}'+{y}={0},{y}{left({0}right)}={1},{y}'{left({0}right)}={1}

fortdefruitI 2020-10-31 Answered
Use the Laplace transform to solve the given initial-value problem
y+2y+y=0,y(0)=1,y(0)=1
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Expert Answer

Isma Jimenez
Answered 2020-11-01 Author has 84 answers
Step 1
It is given that the initial value problem is y+2y+y=0,y(0)=1,y(0)=1
Step 2
Take Laplace transformation as,
L{y}+2L{y}+L{y}=L{0}
s2Y(s)sy(0)y(0)+2[sY(s)y(0)]+Y(s)=0
s2Y(s)sy(0)y(0)+2sY(s)2y(0)+Y(s)=0
s2Y(s)s1+2sY(s)2+Y(s)=0
Y(s)(s2+2s+1)s3=0
Y(s)(s2+2s+1)=s+3
Step 3
On further simplification,
Y(s)=s+3(s+1)2
y(t)=L1{s+3(s+1)2}
y(t)=L1{2(s+1)2+1s+1}       (because  s+3(s+1)2=2(s+1)2+1s+1)
y(t)=L1{2(s+1)2}+L1{1s+1}
y(t)=2tet+et
y(t)=et(2t+1)
Step 4
The solution of the initial value problem is y(t)=et(2t+1)
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