Solve Using Laplace transform frac{dy}{dt}=y+10u_4(t)sin(2(t-4)) , y(0)=2

nagasenaz 2021-01-08 Answered
Solve Using Laplace transform
dydt=y+10u4(t)sin(2(t4)),y(0)=2
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Expert Answer

Roosevelt Houghton
Answered 2021-01-09 Author has 106 answers
Step 1
The given IVP is as follows.
dydt=y+10u4(t)sin(2(t4)),y(0)=2
Apply Laplace transform on both sides of the above equations as follows.
L{dydt}=L{y+10u4(t)sin(2(t4))}
L{dydt}=L{y}+10L{u4(t)sin(2(t4))}
sL{y}y(0)=L{y}+10e4s2s2+4
sL{y}2=L{y}+20e4ss2+4
L{y}(s1)=2+20e4ss2+4
L{y}=2s1+20e4s(s1)(s2+4)
Step 2
Apply inverse Laplace transform on both sides as follows.
L{y}=2s1+20e4s(s1)(s2+4)
L1{L{y}}=L1{2s1+20e4s(s1)(s2+4)}
y(t)=2L1{1s1}+L1{e4s20(s1)(s2+4)}
y(t)=2L1{1s1}+L1{e4s[4s14ss2+44s2+4]}
y(t)=2et+u(t4)[4et44cos(2(t4))2sin(2(t4))]
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