My Math Forum  

Go Back   My Math Forum > College Math Forum > Differential Equations

Differential Equations Ordinary and Partial Differential Equations Math Forum


Reply
 
LinkBack Thread Tools Display Modes
June 16th, 2017, 10:49 AM   #1
Newbie
 
Joined: Jan 2017
From: Costa Rica

Posts: 3
Thanks: 0

How to solve this inverse transform Laplace?

1/(s^2+1)^2

Knowing the Laplace transform of L(sen t-cos t) may help you.
Macoleco is offline  
 
June 16th, 2017, 11:28 AM   #2
Global Moderator
 
Joined: Dec 2006

Posts: 17,919
Thanks: 1383

Knowing the Laplace transform of t*cos(t) is particularly helpful.
skipjack is offline  
June 18th, 2017, 07:58 AM   #3
Newbie
 
Joined: Dec 2016
From: Austin

Posts: 11
Thanks: 1

Solution

Problem:
$\displaystyle \mathcal{L}^{-1} \left \{ \frac{1}{ \left ( s^{2}+1 \right )^{2} } \right \}$

Recall:
$\displaystyle \mathcal{L}^{-1} \left \{ \frac{2a^{3}}{ \left ( s^{2}+a^{2} \right )^{2} } \right \}=\sin \left ( at \right )-at \cos \left ( at \right )$

We can use this definition by making a couple convenient substitutions and algebraic manipulations to make the problem look like our definition.

Let:
$\displaystyle a=1$

Then our problem becomes:
$\displaystyle \mathcal{L}^{-1} \left \{ \frac{a^{3}}{ \left ( s^{2}+a^{2} \right )^{2} } \right \}$

We now need a 2 in the numerator to complete the definition, so multiply top and bottom by 2:

$\displaystyle \mathcal{L}^{-1} \left \{ \frac{2a^{3}}{ 2 \left ( s^{2}+a^{2} \right )^{2} } \right \}$

Factor out a $\displaystyle \frac{1}{2}$:

$\displaystyle \frac{1}{2} \mathcal{L}^{-1} \left \{ \frac{2a^{3}}{ \left ( s^{2}+a^{2} \right )^{2} } \right \}$

We now have the definition, so substitute:
$\displaystyle =\frac{1}{2} \left [ \sin \left ( at \right )-at \cos \left ( at \right ) \right ]$

Finally, substitute $\displaystyle a=1$ back into the solution to get:

$\displaystyle =\frac{1}{2} \left [ \sin \left ( t \right )-t \cos \left ( t \right ) \right ]$

Hope that helps!
thegrade is offline  
Reply

  My Math Forum > College Math Forum > Differential Equations

Tags
inverse, laplace, solve, transform



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Relatioship between Laplace and Inverse Laplace Transform from tables szz Differential Equations 1 November 2nd, 2014 03:18 AM
ODE and Inverse Laplace Transform szz Calculus 1 November 1st, 2014 03:14 PM
Inverse laplace transform Rydog21 Calculus 2 October 15th, 2013 09:35 AM
Laplace tranform and inverse of Laplace transform Deiota Calculus 1 April 28th, 2013 10:28 AM
inverse laplace transform defunktlemon Real Analysis 1 April 14th, 2012 02:19 PM





Copyright © 2017 My Math Forum. All rights reserved.