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Chapter 4Laplace TransformIt’s time to stop guessing solutions and find a systematic way of finding solutionsto non homogeneous linear ODEs. We define the Laplace transform of a functionfin the following way.Definition 4.1.The Laplace transform off(t), writtenF(s), is given by(4.1)L(f) =integraldisplay0estf(t)dt.That is, the Laplace transform acts on a function,f(t), integrates thetout, andcreates function ofs, which we denoteF(s).Before we see why this is useful, we might want to know if the integral in thedefinition exists - it is after all an improper integral. We need another definition.Definition 4.2.We say a functionf(t) is exponentially bounded or of exponentialorder if there exists non negative numbersa,kandMsuch that(4.2)f(t)keattM.We want to allow for interesting functions forcing an ODE.Definition 4.3.A functionfis said to be piecewise continuous on a boundedinterval if it has a finite number of discontinuities and the left and right limits existat each discontinuity. It is said to be piecewise continuous on [0,] if it is piecewisecontinuous on every bounded subintervalI[0,].And finally,Definition 4.4.A functionfis said to be piecewise smooth iffand its derivativeare piecewise continuous.Now we can say when the Laplace transform of a function exists.Theorem 4.5.Supposefis piecewise continuous on[0,]and exponentiallybounded. ThenL(f) =F(s)exists for alls > a.37
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Stop automatic conversion from PDF to Word I am using Win 8.1, 64 bit, office 365 home.personal. When I download a PDF file and save it, the next time I try to open it a conversion from the PDF to word takes place automatically. STOP Procrastinating and Get Things Done Now! Worksheet # 1 – “Why We Procrastinate” Directions. Take your best guess and rank the following reasons for why we procrastinate from 1 to 5 starting with “1” being the biggest reason we procrastinate and “5” being the smallest.

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