Solve the initial value problem below using the method of Laplace transforms 2ty''-3ty'+3y=6,y(0)=2, y'(0)=-3

Emily-Jane Bray 2021-02-09 Answered
Solve the initial value problem below using the method of Laplace transforms
2ty3ty+3y=6,y(0)=2,y(0)=3
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Expert Answer

unett
Answered 2021-02-10 Author has 119 answers
Step 1
Given initial value problem,
2ty3ty+3y=6,y(0)=2,y(0)=3
Step 2
Take Laplace transform of both sides of the given initial value problem,
L[2ty3ty+3y]=L[6]
2L[ty]3L[ty]+3L[y]=6L[1]
Use the formula such that
L[ty]=(1)d(ds)L[y]
L[y]=s2L[y]sy(0)y(0)
L[y]=sL[y]y(0)
L[1]=1s
Step 3
Then,
2[s2L[y]sy(0)y(0)]3[sL[y]y(0)]+3L[y]=61s
2[s2L[y]2s+3]3[sL[y]2]+3L[y]=6s
2s2L[y]4s+63sL[y]+6+3L[y]=6s
(2s23s+3)L[y]4s+12=6s
(2s23s+3)L[y]=6s+4s12
L[y]=6+4s212s2s23s+3
L[y]=6+4s212s2s23s+3
Step 4
Taking inverse Laplace transform,
y=L1[6+4s212s2s23s+3](1)
Now
6+4s212s2s23s+3=26s2s23s+3
=23ss232s+32
=23(s34+34)(s34)2+1516
=23(s34)(s34)2+1516941(s34)2+1516
Step 5
Then,
L1[6+4s212s2s23s+3]=L1[23(s34)(s34)2+1516941(s34)2+1516]

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