TopicsSections 7.1 to 7.6 The textbook topics, definitions, examples and theorems

Edwards-Penney Ch 3, 7.1 to 7.5 (20.1 K, txt, 17 Apr 2015)

Forward table Backward tableBasic Theorems of Laplace TheoryFunctions of exponential order Existence theorem for Laplace integrals Euler solution atoms have a Laplace integral Forward table Backward table Lerch's theorem Linearity. The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)). Shift theorem L(exp(at)f(t)) = L(f(t))|s->(s-a) Parts theorem L(y')=sL(y)-y(0) Parts formula derivation. Solving differential equations by Laplace's method. Slide: Solving y' = -1, y(0)=2 with Laplace's method Laplace's method and quadrature for higher order equations and systems Solving x'' + 4x = t exp(-t), x(0)=1, x'(0)=0 by the Laplace methodLaplace theory references: Laplace and Newton calculus. Photos. (200.2 K, pdf, 03 Mar 2012)Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 18 Mar 2012)Slides: Laplace rules (160.3 K, pdf, 03 Mar 2012)Slides: Laplace table proofs (169.6 K, pdf, 03 Mar 2012)Slides: Laplace examples (149.1 K, pdf, 03 Mar 2012)Slides: Piecewise functions and Laplace theory (108.5 K, pdf, 03 Mar 2013)Slides: DE systems, examples, theory (730.9 K, pdf, 10 Apr 2014)Manuscript: Laplace resolvent method (88.1 K, pdf, 03 Mar 2012)Slides: Laplace second order systems (288.1 K, pdf, 03 Mar 2012)Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 09 Apr 2014)Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008)Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008)Slides: Heaviside's method 2008 (352.3 K, pdf, 07 Jan 2014)Manuscript: Laplace theory 2008 (500.9 K, pdf, 16 Mar 2014)Manuscript: Ch7 Laplace solutions 7.1 to 7.4 (from EP 2250 book) (1068.7 K, pdf, 22 Feb 2015)Transparencies

DEF:Piecewise Continuous FunctionExistence of the Laplace integral. One-sided and two-sided Laplace integral Freeway example, suspension collides with a ramp. DEF. Gamma function Gamma(t) = integral x=0 to x=infinity x^{t-1} e^{-x} Gamma(n)=(n-1)!, generalizes the factorial function DEF. Mellin transform {Mf}(s)= phi(s)=integral x=0 to x=infinity x^{s-1} f(x) DEF. Two-sided Laplace transform {Bf}(s) = {Mf(-ln(x))}(s) = integral x=0 to x=infinity x^{s-1}f(-ln x) DEF. Unit step u(t-a)=1 for t>=a, else zero DEF. Ramp t->(t-a)u(t-a) Backward table problems: examples Forward table problems: examples Computing Laplace integrals L(f(t)) with rules Solving an equation L(y(t))=expression in s for y(t) Complex roots and quadratic factors Partial fraction methods Trig identities and their use in Laplace calculations Hyperbolic functions and Laplace calculations Why the forward and backward tables don't have cosh, sinh entries

Piecewise FunctionsUnit Step: u(t)=1 for t>=0, u(t)=0 for t<0. Pulse: pulse(t,a,b)=u(t-a)-u(t-b) Ramp: ramp(t-a)=(t-a)u(t-a) L(u(t-a)) = (1/s) exp(-as) [for a >= 0 only]Integral TheoremL(int(g(x),x=0..t)) = s L(g(t)) Applications to computing ramp(t-a) L(ramp(t-a)) = (1/s^2) exp(-as) [for a >= 0 only]Piecewise defined periodic wavesSquare wave: f(t)=1 on [0,1), f(t)=-1 on [1,2), 2-periodic Triangular wave: f(t)=|t| on [-1,1], 2-periodic Sawtooth wave: f(t)=t on [0,1], 1-periodic Rectified sine: f(t)=|sin(kt)| Half-wave rectified sine: f(t)=sin(kt) when positive, else zero. Parabolic wavePeriodic function theoremProof details Laplace of the square wave. Problem 7.5-25. Answer: (1/s)tanh(as/2) Applications of Laplace's method from 7.3, 7.4, 7.5Convolution theoremDEF. Convolution of f and g = f*g(t) = integral of f(x)g(t-x) from x=0 to x=t THEOREM. L(f(t))L(g(t))=L(convolution of f and g) Application: L(cos t)L(sin t) = L(0.5 t sin(t))Second shifting Theoremse^{-as}L(f(t))=L(f(t-a)u(t-a)) Backward table L(g(t)u(t-a))=e^{-as}L(g(t+a)) Forward table EXAMPLES. Forward table L(sin(t)u(t-Pi)) = e^{-Pi s} L(sin(t)|t->t+Pi) = e^{-Pi s} L(sin(t+Pi)) = e^{-Pi s} L(sin(t)cos(Pi)+sin(Pi)cos(t)) = e^{-Pi s} L(-sin(t)) = e^{-Pi s} ( -1/(s^2+1)) Backward table L(f(t)) = e^{-2s}/s^2 = e^{-2s} L(t) = L(t u(t)|t->t-2) = L((t-2)u(t-2)) Therefore f(t) = (t-2)u(t-2) = ramp at t=2.Laplace Resolvent Method.==> This method is a shortcut for solving systems by Laplace's method. ==> It is also a convenient way to solve systems with maple.: Laplace resolvent method (88.1 K, pdf, 03 Mar 2012)Slides

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