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# LinearAnalysis MATH287

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This 3 page Class Notes was uploaded by Leopoldo Rutherford on Sunday October 11, 2015. The Class Notes belongs to MATH287 at Cuesta College taught by RobertSchwennicke in Fall. Since its upload, it has received 37 views. For similar materials see /class/221241/math287-cuesta-college in Mathematics (M) at Cuesta College.

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Date Created: 10/11/15

Math 287 sections 41 46 Vocabulary 0 Characteristic equation 0 Phase plane 0 Characteristic roots 0 Transient solution 0 Multiplicity of a root 0 Steady State solution 0 Overdamped critically damped o Beats underdamped o Resonance 0 Basis Fundamental Set 0 Quasi Frequency Period 0 Coset 0 Time Varying Amplitude 0 General solution Problems to be Solved 0 Second order homogeneous linear differential equations and IVPs with constant coefficients 0 Second order nonhomogeneous linear differential equations and IVPs with constant coefficients using the method of undetermined coefficients w the method of variation of parameters 0 Higher order homogeneous linear differential equations with constant coefficients 0 Word problem involving a massspring system 0 Problem involving an LRC 7 Circuit given the differential equation Other Computations and things to do Pro 0 Converting to the altematepreferred single cosine form for the solution of an IVP 0 Determining the transient and steady state solutions of an IVP o Calculating the resonant frequency of an LRCCircuit or of a massspring system 0 The quasi period and time varying amplitude of the solution to an IVP o Recognize the phase plane of over critically damped motion underdamped motion forced motion 0 Draw a trajectory in the phase plane for the solution to an IVP o Recognize the graph of overcritically damped motion underdamped motion forced motion 0 Interpreting the coefficients of a massspring differential equation 0 Analyzing the solution to a massspring system solution ofs and Demonstrations o The solution set of a second order linear homogeneous differential equation is two dimensional page 217 o Distinct characteristic roots produce linearly independent solutions 42 44 o If r is a repeated characteristic root then 6quot and requot are linearly independent 42 46 o The mass in an overdamped or critically damped system can cross the equilibrium point at most once 42 51 52 Math 287 sections 81 84 Formulas and techniques you need to know 0 Completing the Square Partial Fraction Decomposition The sum of an in nite geometric Series The convolution of two functions 000 Principles for Linear Differential Equations 0 Homogeneous Super Position Principle 0 Nonhomogeneous Principle 0 Nonhomogeneous Super Position Principle Com putations 0 Calculate the Laplace transform of a function using the de nition 0 Laplace transform and inverse Laplace transform using the formulas on page 472 o Laplace transform and inverse Laplace transform for functions involving the unit step function or the unit impulseDirac delta function 0 Laplace transform of a periodic function 0 Solve IVP or a differential equation using the Laplace transform 0 Calculate a convolution 0 Write the solution to an IVP or a DE in terms of a convolution of the impulse response function Proofs Derivations 0 Any formula from the table on page 472 o Linearity of the Laplace transform and the inverse Laplace transform 0 The derivative theorem for Laplace transform 476 o The Delay Theorem 487 488 0 Periodic Formula 499 Lsinbt2L Lcosbt 2 LftPLft Table of Laplace Transforms no L ft Fm j e39 ft dr 0 1 a b Le smbtm n a s a Lt 1 n012 8 L e cosbt W 1 b L t 9 L 39nhb e Sa 1 I 32 b2 Ltquot e 39n 10 Lcoshbt S San1 S2b2 d S 132 Ltft EFS S Le ft Fs a s b2 Lf39t 3FSf0 L f t s2 Fm s f0 f 0 Lf quot r squotFs s 391f0 f 0 e L stept 61 L t1 S Lft astept a e Fs 21 L6t a e L ftstept a yam ft a P 1 lisp I f 0dr periodic lnction e 0

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