Solve both a)using the integral definition , find the convolution f∗g of f(t)=cos⁡2t,g(t)=et b) Using above answer , find the Laplace Transform of f*g

Name: Solve both a)using the integral definition , find the convolution f∗g of f(t)=cos⁡2t,g(t)=et b) Using above answer , find the Laplace Transform of f*g
Uploaded: 2022-02-23T17:22:57+00:00
Description: Solve both a)using the integral definition , find the convolution f∗g of f(t)=cos⁡2t,g(t)=et b) Using above answer , find the Laplace Transform of f*g

Question

Solve both  a)using the integral definition , find the convolution  f∗g of f(t)=cos⁡2t,g(t)=et  b) Using above answer , find the Laplace Transform of f*g

d2saint0 · Accepted Answer

Step 1  Given:  The functions f(t)=cos⁡2t,g(t)=et  Step 2   a) Definition of convolution:  The convolution of piecewise continuous functions f, g : R→R is the function f * g : R→R given by:  (f∗g)(t)=∫0tf(τ)g(t−τ)dτ  Therefore, by definition  (f∗g)(t)=∫0tcos⁡(2τ)et−τdτ Integrate using integration by parts:  u=cos⁡(2τ),dv=e(t−τ)dτ  du=−2sin⁡(2τ)dτ,v=∫et−τdτ=−et−τ∫udv=uv−∫vdu  ⇒I=∫cos⁡(2τ)et−τdτ=−cos(2τ)et−τ−∫(−et−τ)(−2sin⁡(2τ)dτ)  ⇒I=−cos⁡(2τ)et−τ−2∫(et−τ)(sin⁡(2τ))dτ…(1)  Step 3  To simplify further, use parts of integration for the second term Compute the integral  ∫et−τsin⁡(2τ)dτ  u=sin⁡(2τ),dv=et−τdτ  du=2cos⁡(2τ),v=−et−τ  ⇒∫et−τsin⁡(2τ)dτ=−et−τsin⁡(2τ)−∫(−et−τ)(2cos⁡(2τ))dτ  =−et−τsin⁡(2τ)+2∫(et−τ)(cos⁡(2τ))dτ  ⇒∫et−τsin⁡(2τ)dτ=−et−τsin⁡(2τ)+2I  {Because I=∫(et−τ)(cos⁡(2τ))dτ}  Equation (1) becomes:  ⇒I=−cos⁡(2τ)et−τ−2[−et−τsin⁡(2τ)+2I]  ⇒I=−cos⁡(2τ)et−τ+2et−τsin⁡(2τ)−4I  ⇒5I=−cos⁡(2τ)et−τ+2et−τsin⁡(2τ)  ⇒I=−15cos⁡(2τ)et−τ+25et−τsin⁡(2τ)  ∫cos⁡(2τ)et−τdτ=−15cos⁡(2τ)et−τ+25et−τsin⁡(2τ)  Step 4  Substitute the upper and lower limit and simplify

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

Answered question

Answer & Explanation

New Questions in Differential Equations