Use Laplace transforms to solve the initial value problems. y"+y'=\delta(t)-\delta(t-3), y(0)=1 , y'(0)=0

Tazmin Horton

Tazmin Horton

Answered question

2021-09-18

Use Laplace transforms to solve the initial value problems.
yy=δ(t)δ(t3),y(0)=1,y(0)=0

Answer & Explanation

toroztatG

toroztatG

Skilled2021-09-19Added 98 answers

L{δ(ta)}=eas
L1{1sa}=eat
L1{eass}=u(ta)
Given
yy=δ(t)δ(t3),y(0)=1,y(0)=0
Taking laplace transformation both sides:
L{y"}+L{y}=L{δ(t)}L{δ(t3)}
[s2Y(s)sy(0)y(0)]+[sy(s)y(0)]=1e3s
(s2y(s)s)+(sy(s)1)=1e3s
y(s)[s2+s]s1=1e3s
y(s)[s2+s]=2e3s+s
y(s)=2+se3ss(s+1)
y(s)=2+ss(s+1)e3ss(s+1)
y(s)=2s1s+1e3ss+e3ss+1
Now taking inverse laplace transformation both sides:
L1{y(s)}=2L1{1s}L1{1s+1}L1{e3ss}+L1{e3ss+1}
y(t)=2×1etu(t3)+e3tu(t3)
y(t)=2et+(e

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