Solve Using Laplace transform dydt=y+10u4(t)sin⁡(2(t−4)),y(0)=2

Name: Solve Using Laplace transform dydt=y+10u4(t)sin⁡(2(t−4)),y(0)=2
Uploaded: 2022-02-23T17:22:57+00:00
Description: Solve Using Laplace transform dydt=y+10u4(t)sin⁡(2(t−4)),y(0)=2

Question

Solve Using Laplace transform  dydt=y+10u4(t)sin⁡(2(t−4)),y(0)=2

Roosevelt Houghton · Accepted Answer

Step 1  The given IVP is as follows.  dydt=y+10u4(t)sin⁡(2(t−4)),y(0)=2  Apply Laplace transform on both sides of the above equations as follows.  L{dydt}=L{y+10u4(t)sin⁡(2(t−4))}  L{dydt}=L{y}+10L{u4(t)sin⁡(2(t−4))}  sL{y}−y(0)=L{y}+10e−4s2s2+4  sL{y}−2=L{y}+20e−4ss2+4  L{y}(s−1)=2+20e−4ss2+4  L{y}=2s−1+20e−4s(s−1)(s2+4)  Step 2  Apply inverse Laplace transform on both sides as follows.  L{y}=2s−1+20e−4s(s−1)(s2+4)  L−1{L{y}}=L−1{2s−1+20e−4s(s−1)(s2+4)}  y(t)=2L−1{1s−1}+L−1{e−4s20(s−1)(s2+4)}  y(t)=2L−1{1s−1}+L−1{e−4s[4s−1−4ss2+4−4s2+4]}  y(t)=2et+u(t−4)[4et−4−4cos⁡(2(t−4))−2sin⁡(2(t−4))]

Solve Using Laplace transform dydt=y+10u4(t)sin⁡(2(t−4)),y(0)=2

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