November 1st, 2014, 01:58 PM  #1 
Senior Member Joined: Oct 2014 From: EU Posts: 224 Thanks: 26 Math Focus: Calculus  ODE and Inverse Laplace Transform
Hi all, I have a question related to an exercise regarding the resolution of ODE with the Inverse Laplace Transform. I have a worked example as such: "Use Laplace transforms to solve the differential equation" $\displaystyle 2 {\text{d}^2y \over \text{d}x^2} + 5 {\text{d}y \over \text{d}x}  3y = 0 $ "given when $\displaystyle y(0) = 4$ and $\displaystyle y'(0) = 9$" The the resolution starts: $\displaystyle \begin{aligned} & 2 \mathcal{L} \left [ {\text{d}^2y \over \text{d}x^2} \right ] + 5 \mathcal{L} \left [ {\text{d}y \over \text{d}x} \right ]  3 \mathcal{L} \left [ y \right ] = \\ & 2 ( s^2 \mathcal{L} [y] sy(0)  y'(0) ) + 5(s\mathcal{L} [y]  y(0) ) 3 \mathcal{L} [y] = \\ & \cdots \end{aligned} $ And then the ODE is solved solving for $\displaystyle \mathcal{L} [y]$. If I am not wrong, given and ODE of the order $\displaystyle n$, we always apply the operator $\displaystyle s^n \mathcal{L} [y]$ where again, $\displaystyle n$ is the order of the differential equation, and then the same coefficient $\displaystyle n$ gets smaller as we evaluate de initial conditions.. So, for example, given a third order differential equation: $\displaystyle 6 {\text{d}^3y \over \text{d}x^3} = 0 $ with the initial conditions: $\displaystyle \begin{aligned} & y(0) = 2 \\ & y'(0) = 5 \\ & y''(0) = 3 \\ \end{aligned}$ I should proceed as follows: $\displaystyle \begin{aligned} & 6 ( s^3 \mathcal{L} [y]  s^2y(0)  sy'(0)  y''(0) ) = \\ & 6 ( s^3 \mathcal{L} [y]  2s^2  5s  3 ) = \\ & {2 \over s} + {5 \over s^2} + {3 \over s^3} \end{aligned} $ Am I right ? Thank you in advance. Last edited by szz; November 1st, 2014 at 02:56 PM. 
November 1st, 2014, 03:14 PM  #2 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,889 Thanks: 768 Math Focus: Wibbly wobbly timeywimey stuff. 
Looks good to me. Dan 

Tags 
inverse, laplace, ode, transform 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Inverse Laplace Transform  Bkalma  Calculus  1  May 6th, 2014 07:08 PM 
Inverse laplace transform  Rydog21  Calculus  2  October 15th, 2013 09:35 AM 
inverse laplace transform  capea  Complex Analysis  5  August 3rd, 2013 02:30 PM 
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 