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 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. June 16th, 2017, 11:28 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,973 Thanks: 2224 Knowing the Laplace transform of t*cos(t) is particularly helpful. June 18th, 2017, 07:58 AM #3 Newbie   Joined: Dec 2016 From: Austin Posts: 16 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! Tags inverse, laplace, solve, transform Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post szz Differential Equations 1 November 2nd, 2014 03:18 AM szz Calculus 1 November 1st, 2014 03:14 PM Rydog21 Calculus 2 October 15th, 2013 09:35 AM Deiota Calculus 1 April 28th, 2013 10:28 AM defunktlemon Real Analysis 1 April 14th, 2012 02:19 PM

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