Solve the following IVP using Laplace Transform y'-2y =1-t , y(0)=4

smileycellist2 2020-11-01 Answered
Solve the following IVP using Laplace Transform
y2y=1t,y(0)=4
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Expert Answer

berggansS
Answered 2020-11-02 Author has 91 answers
Step 1
The Laplace transform is given as:
y2y=1t,y(0)=4
Taking the laplace trasform on both sides:
L{y}2L{y}=L(1)L(t)
sL(y)y(0)2L(y)=1s1s2
sL(y)42L(y)=s1s2
L(y)(s2)=s1s2+4
L(y)=4s2+s1s2(s2)
Step 2
Taking the inverse Laplace transform and make into the fraction form:
(y)=L1{4s2+s1s2(s2)}(i)
4s2+s1s2(s2)=As+Bs2+C(s2)
4s2+s1=As(s2)+B(s2)+Cs2
4s2+s1=(A+C)s2+(2A+B)s2B
Step 3
Compare the coefficient and find out the values of A, B and C:
4s2+s1=(A+C)s2+(2A+B)s2BA+C=4,2A+B=1 and 2B=1B=12
2A+B=12A=112A=14
A+C=4C=4(14)C=174
4s2+s1s2(s2)=14s+12s2+174(s2)
Step 4
Now, form equation (i):
(y)=L1{4s2+s1s2(s2)}=L1{14s+12s2+174s2}
y=14L1{1s}+12L1{1s2}+174L1{1s2}
Using L1{1s}=(1),L1{1s2}=t,L1{1sa}=eat
y=14+t2+174e2t
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