Solve the equation.delta(t-t_0) is the Dirac-Delta function. y"+4y'+5y=delta(t-2pi)+delta(t-4pi) , y(0)=0 , y'(0)=0

Brittney Lord 2020-12-05 Answered
Solve the equation.δ(tt0) is the Dirac-Delta function.
y"+4y+5y=δ(t2π)+δ(t4π) , y(0)=0 , y(0)=0
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Expert Answer

Dora
Answered 2020-12-06 Author has 98 answers
Step 1
Consider the given IVP as follows.
y"+4y+5y=δ(t2π)+δ(t4π) , y(0)=0 , y(0)=0
Apply Laplace transform on both sides as follows.
L{y"+4y+5y}=L{δ(t2π)+δ(t4π)}
L{y"}+4L{y}+5L{y}=L{δ(t2π)}+L{δ(t4π)}
s2L{y}sy(0)y(0)+4sL{y}4y(0)+5L{y}=e2πs+e4πs
s2L{y}+4sL{y}+5L{y}=e2πs+e4πs
L{y}[s2+4s+5]=e2πs+e4πs
L{y}=e2πs+e4πss2+4s+5
L{y}=e2πs(s+2)2+1+e4πs(s+2)2+1
Step 2
Apply inverse Laplace transform to solve the given IVP as follows.
L{y}=e2πs(s+2)2+1+e4πs(s+2)2+1
y(t)=L1{e2πs(s+2)2+1+e4πs(s+2)2+1}
=L1{e2πs(s+2)2+1}+L1{e4πs(s+2)2+1}
=u(t2π)e2(t2π)sin(t2π)+u(t4π)e2(t4π)sin(t4π)
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