Diffeq litercy

# Diffeq litercy - Differential Equations&amp;Mathematica...

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Mathematica Authors: Bruce Carpenter, Bill Davis and Jerry Uhl ©2001-2007 Publisher: Math Everywhere, Inc. Version 6.0 DE.02 Transition from Calculus to DiffEq: The Forced Oscillator DiffEq y ! @ t D + by ¢ @ t D + cy @ t D = f @ t D LITERACY What you need to know when you're away from the machine. · L.1) Here are plots of two unforced oscillators: 5 10 15 20 t - 1.5 - 1.0 - 0.5 0.5 1.0 1.5 2.0 y @ t D 5 10 15 20 t - 1 1 2 y @ t D One of these plots is the solution of y ££ @ t D + 0.2 y £ @ t D + 5.1 y @ t D = 0 with y @ 0 D = 1.5 and y £ @ 0 D = 3.0. The other is a plot of the solution of y ££ @ t D + 0.4 y £ @ t D + 5.1 y @ t D = 0; with y @ 0 D = 1.5 and y £ @ 0 D = 3.0. Which is which? L.2) Here are plots of the solutions of diffeq1): y ££ @ t D + 4.7 y @ t D = 0 diffeq2): y ££ @ t D + 0.3 y £ @ t D + 4.7 y @ t D = 0, diffeq3): y ££ @ t D + 1.3 y £ @ t D + 4.7 y @ t D = 0, diffeq4): y ££ @ t D + 7.3 y £ @ t D + 4.7 y @ t D = 0, all with the same starter data y @ 0 D = 2 and y £ @ 0 D = 3. The plots are not in order. Your job is to match the differential equation with the plot of its solution. 2 4 6 8 t - 2 - 1 1 2 Plot A 2 4 6 8 t - 2 - 1 1 2 Plot B 2 4 6 8 t - 2 - 1 1 2 Plot C 2 4 6 8 t - 2 - 1 1 2 Plot D diffeq1) ö Plot. ....... diffeq2) ö Plot. ....... diffeq3) ö Plot. ....... diffeq4) ö Plot. ....... How does the starter data signal that each plot must go up before it can go down? L.3) a) Write down the characteristic equation for this unforced linear oscillator diffeq y ££ @ t D + 6 y £ @ t D + 25 y @ t D = 0 b) Solve the characteristic equation. c) What information about all solutions of y ££ @ t D + 6 y £ @ t D + 25 y @ t D = 0 do you get by inspecting the solutions of the characteristic equation? · L.4) a) Write down the characteristic equation for this unforced linear oscillator diffeq y ££ @ t D + 7 y £ @ t D + 12 y @ t D = 0 b) Solve the characteristic equation. c) Explain how the solutions of the characteristic equation signal take all solutions of y ££ @ t D + 7 y £ @ t D + 12 y @ t D = 0 head to 0 without oscillation. · L.5) a) Write down the characteristic equation for this unforced linear oscillator diffeq y ££ @ t D + 7 y £ @ t D + 25 y @ t D = 0. b) What information about all solutions of y ££ @ t D + 7 y £ @ t D + 25 y @ t D = 0 do you get when you solve the characteristic equation? L.6) Here are three plots of solutions of a damped forced oscillator y ££ @ t D + 0.7 y £ @ t D + 5.5 y @ t D = 2 Sin @ 3 t D + Cos @ 1.5 t D . Two of the solutions have random starting values on y @ 0 D and y £ @ 0 D . The other solution is yzeroinput

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## This note was uploaded on 09/21/2011 for the course MATH 285 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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Diffeq litercy - Differential Equations&amp;Mathematica...

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