Project Four December 9, 2008
I did a fair amount of reading to be able to better understand LaPlace Transforms to get an idea of why and when they are used. The formulas looked rather daunting at first, but as I read through a couple of sections in different books (the two texts recommended for the class and Schaum’s Outlines, and the web page S.O.S. Math – I got a better sense of what was happening theoretically not only with using LaPlace Transforms, but with varies terms and definitions – some that I have worked with but, some others were new to me. Since Matt ( my project partner ) so graciously and quickly caught me up to speed on what I had missed – I felt like I needed to kick into gear very quickly to be a better support to him. It’s nicer to keep things in balance. I read through the class info. that was provided to us on project four and the project guide #4.
I began by writing down words and phrases that I didn’t fully understand. I also wanted to make sure what I thought I knew – I really did know. I began asking myself questions like – why were we given these equations rather than some others – what was their significance to differential equations – what was important about them? What would solving them reveal? Where there any similarities between them? Or did each equation represent or teach us something different? Then as read, I realized that I could also compare these equations to those in previous projects – to talk about similarities and differences not only in the equations themselves – but, the methods used to solve them – why one method was better to use than another – things to look for so that I can eliminate methods that won’t work for example LaPlace Transformations would be used for linear differential equations that are exponentially bounded. The difference between an explicit solution and a general solution – which brought me back to describing things three ways – analytically, qualitatively and numerically. I had asked myself during the first lecture for this project – are there other uses for LaPlace transforms?
As I read – I began to see some connections between Differential Equations, Linear Algebra and LaPlace Transforms – I read about different functions that I had heard about - but never really quite understood really - I began to question what kind of functions we had – just a brief list (not conclusive) functions like the Gamma function, Beta function, Dirac function, Sterling’s formula … etc. As I read about phase planes and damped and undamped harmonic oscillators – then I saw spiral sinks, straight line solutions and equilibrium points – everything that seemed so disjuncted started to not seem so jumbled and disconnencted. I might have to draw a mapping of how it all connects to see it visually in order to communicate it effectively to Matt so that we can discuss each equation in class. I am hoping also to be able to answer the extra credit portion - LaPlace Transforms can be used to solve systems of linear differential equations - sets of two or more differential equations – with an equal number of equations and unknowns. For each differential equation in the system – the LaPlace transform of the unknown functions is taken – you get a set of equations – then the inverse transform is calculated – and you get the explicit answer. The solution seems to use Linear Algebra with the LaPlace Transform … so, a lot to discuss with Matt – better get drawing and organize my thoughts for class. TTFN
