Q57P - Mathematical Methods in Physical Sciences

Question

Find the inverse Laplace transform of the following functions by using (7.16). $\frac{\mathbf{p}\mathbf{+}\mathbf{1}}{\mathbf{p}\left(p^2+1\right)}$

Step-by-Step Solution

Answer

The residues at poles,   $\mathit{f}\left(t\right)\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{1}\mathbf{ }\mathbf{+}\mathbf{ }\mathit{s}\mathit{i}\mathit{n}\mathbf{ }\mathit{t}\mathbf{ }\mathbf{-}\mathbf{ }\mathit{c}\mathit{o}\mathit{s}\mathbf{ }\mathit{t}$

Step 1: To find the poles of by denominator.

Using convolution of    we have to find the inverse transform.

 

The given equation is,

  

 

 

Rewrite it equation as,

 

  

 

To find the poles of   by denominator as,

 

  

 

Simple poles have the above equation,   and