Solve both a)using the integral definition , find the convolution f*g text{ of } f(t)=cos 2t , g(t)=e^t b) Using above answer , find the Laplace Transform of f*g

Isa Trevino 2020-10-28 Answered
Solve both
a)using the integral definition , find the convolution
fg of f(t)=cos2t,g(t)=et
b) Using above answer , find the Laplace Transform of f*g
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Expert Answer

d2saint0
Answered 2020-10-29 Author has 89 answers
Step 1
Given:
The functions f(t)=cos2t,g(t)=et
Step 2
a) Definition of convolution:
The convolution of piecewise continuous functions f, g : RR is the function f * g : RR given by:
(fg)(t)=0tf(τ)g(tτ)dτ
Therefore, by definition
(fg)(t)=0tcos(2τ)etτdτ Integrate using integration by parts:
u=cos(2τ),dv=e(tτ)dτ
du=2sin(2τ)dτ,v=etτdτ=etτudv=uvvdu
I=cos(2τ)etτdτ=cos(2τ)etτ(etτ)(2sin(2τ)dτ)
I=cos(2τ)etτ2(etτ)(sin(2τ))dτ(1)
Step 3
To simplify further, use parts of integration for the second term Compute the integral
etτsin(2τ)dτ
u=sin(2τ),dv=etτdτ
du=2cos(2τ),v=etτ
etτsin(2τ)dτ=etτsin(2τ)(etτ)(2cos(2τ))dτ
=etτsin(2τ)+2(etτ)(cos(2τ))dτ
etτsin(2τ)dτ=etτsin(2τ)+2I
{Because I=(etτ)(cos(2τ))dτ}
Equation (1) becomes:
I=cos(2τ)etτ2[etτsin(2τ)+2I]
I=cos(2τ)etτ+2etτsin(2τ)4I
5I=cos(2τ)etτ+2etτsin(2τ)
I=15cos(2τ)etτ+25etτsin(2τ)
cos(2τ)etτdτ=15cos(2τ)etτ+25etτsin(2τ)
Step 4
Substitute the upper and lower limit and simplify

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