Find the solution of the following differential equation by Laplace transforms: y'''- 5y" + 7y’-3y =20sin(t) , y(0)=y'(0)=0 , y"(0)=-2

geduiwelh 2020-12-15 Answered
Find the solution of the following differential equation by Laplace transforms:
y5y"+7y3y=20sin(t),y(0)=y(0)=0,y"(0)=2
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Expert Answer

Isma Jimenez
Answered 2020-12-16 Author has 84 answers
Step 1
Consider the following initial value problem and apply the Laplace transform on both sides:
y5y"+7y3y=20sint
y(0)=y(0)=0,y"(0)=2
s3L{y(t)}s2y(0)sy(0)y"(0)5{s2L{y(t)}sy(0)y(0)}+7{sL{y(t)}y(0)}3L{y(t)}=20(s2+1)
s3Y{y(t)}+25{s2L{y(t)}}+7{sL{y(t)}}3L{y(t)}=20(s2+1)
(s35s2+7s3)L{y(t)}+2=20(s2+1)
(s35s2+7s3)L{y(t)}=20(s2+1)2
L{y(t)}=20(s2+1)(s35s2+7s3)2(s35s2+7s3)
=13s(s2+1)+3(s+1)4(s1)2
=1(s2+1)+3(s+1)4(s1)23s(s2+1)
Step 2
Apply the inverse Laplace transform:
L{y(t)}=1(s2+1)+3(s+1)4(s1)23s(s2+1)
{y(t)}=L1[1(s2+1)+3(s+1)4(s1)23s(s2+1)]
y(t)=sint+3et4tet3cost
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