Find the equation by applying the Laplace transform. displaystyle{y}^{{{left({4}right)}}}-{y}= sin{{h}}{t} y(0)=y'(0)=y"(0)=0 y'''(0)=1

melodykap 2021-02-10 Answered
Find the equation by applying the Laplace transform.
y(4)y=sinht
y(0)=y(0)=y"(0)=0
y(0)=1
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Expert Answer

avortarF
Answered 2021-02-11 Author has 113 answers
Step 1
The given equation is,
y4y=sinht(1)
The given initial condition is,
y(0)=1(2)
y(0)=1(3)
y(0)=1(4)
y(0)=1(5)
The Laplace transform is given as,
L{Fn(t)}=snL{F(t)}sn1F(0)sn2F(0)Fn1(0)
L{sinhat}=as2a2
Step 2
The given equation (1) is,
y4y=sinht
Taking Laplace transform on both sides,
L{yy}=L{sinht}
s4L{y}s3y(0)s2y(0)sy(0)y(0)L{y}=1s21
On putting the values of equation (2), (3), (4) & (5) in the above equation,
s4L{y}s3(1)s2(1)s(1)(1)L{y}=1s21
s4L{y}s3s2s1L{y}=1s21
s4L{y}L{y}=1s21+s3+s2+s+1
(s41)L{y}=1+s5s3+s4s2+s3s+s21s21
(s41)L{y}=s5+s4ss21

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