Find the Laplace transform Y(s), of the solution of the IVP y"+3y'+2y=cos(2t) y(0)=0 y'(0)=1 Do not solve the IVP

lwfrgin 2020-12-17 Answered
Find the Laplace transform Y(s), of the solution of the IVP
y"+3y+2y=cos(2t)
y(0)=0
y(0)=1
Do not solve the IVP
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Expert Answer

pattererX
Answered 2020-12-18 Author has 95 answers
Step 1
The given initial value problem is y+3y+2y=cos(2t),y(0)=0,y(0)=1
Find the Laplace transform Y(s) of the solution of the IVP as follows.
Step 2Consider the differential equation y+3y+2y=cos(2t)
Take Laplace transform on both sides and obtain,
L{y"+3y+2y}=L{cos(2t)}
L{y"}+3L{y}+2L{y}=L{cos(2t)}
s2Y(s)sy(0)y(0)+3(sY(s)y(0))+2(Y(s))=s(s2+4)
s2Y(s)sy(0)1+3(sY(s)y(0))+2(Y(s))=s(s2+4)
s2Y(s)1+3sY(s)+2Y(s)=s(s2+4)
Y(s)(s2+3s+2)1=s(s2+4)
Y(s)(s2+3s+2)=s(s2+4)+1
Y(s)=s+(s2+4)(s2+4)(s2+3s+2)
Y(s)=(s2+s+4)(s2+4)(s2+3s+2)
Step 3
Therefore, the Laplace transform Y(s) of the solution of the IVP is,Y(s)=(s2+s+4)(s2+4)(s2+3s+2)
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