DE.01.4.Literacy - Differential Equations&Mathematica 6...

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Unformatted text preview: Differential Equations&Mathematica 6 Authors: Bruce Carpenter, Bill Davis and Jerry Uhl ©2001-2007 4 Publisher: Math Everywhere, Inc. Version 6.0 2 DE.01 Transition from Calculus to DiffEq: The Exponential Differential Equation y¢ @tD + r [email protected] = f @tD LITERACY What you need to know when you're away from the machine. · L.1) Here are three diffeqs: a) y£ @tD + 0.2 [email protected] = 0 b) y£ @tD - 0.2 [email protected] = 0 c) y£ @tD + 0.1 [email protected] = 0. Which of these diffeqs is solved by [email protected] = 13 E-0.2 t ? · L.2) Here are three diffeqs: a) y£ @tD + [email protected] = E2 t b) y£ @tD - [email protected] = E2 t c) y£ @tD = E2 t . Which of these diffeqs is solved by [email protected] = 4 Et + E2 t ? · L.3) Here are three diffeqs: a) y£ @tD + 2 t [email protected] = 0 b) y£ @tD - 2 t [email protected] = 0 c) y£ @tD + [email protected] = 0. Which of these diffeqs is solved by 2 [email protected] = 8 E-t ? · L.4) Here are three diffeqs: a) y'' @tD + 4 [email protected] = 0 b) y'' @tD + 9 [email protected] = 0 c) y'' @tD + 16 [email protected] = 0. Which of these diffeqs is solved by [email protected] = 5 [email protected] tD ? · L.5) Just to see that you can handle yourself away from the machine, come up with formulas for the solutions of the following diffeqs: a) y£ @tD + 0.3 [email protected] = 5 E-t with [email protected] = 5. b) y£ @tD - 0.3 [email protected] = 5 with [email protected] = 1. c) y£ @tD + [email protected] = [email protected] - 3D with [email protected] = 2. · L.6) You know that y£ @tD = a [email protected] for all t's and you know that a is positive and that [email protected] is positive. Does [email protected] go up or down as t advances from left to right? · L.7) If a is negative , does the solution of y£ @tD = a [email protected] with [email protected] = 10 go up or down as t advances from left to right. Can the solution ever go negative? Why or why not? · L.8) The solution y[t] of y£ @tD + r [email protected] = f @tD with [email protected] = starter is given by t [email protected] = E-r t starter + E-r t Ÿ0 Er s f @sD „ s. Explain this: If r > 0, then, no matter what starter is, the plot of y[t] will eventually merge with the plot of t [email protected] = E-r t Ÿ0 Er s f @sD „ s. · L.9) Here are three plots of solutions of a certain forced exponential diffeq: 5 10 15 20 t -2 -4 -6 You are given that all three plots are all either solutions of a) y£ @tD + 0.6 [email protected] = 4 [email protected] tD + [email protected] tD or solutions of b) y£ @tD - 0.6 [email protected] = 4 [email protected] tD + [email protected] tD. Make your choice and say how you arrived at it. · L.10) The solution y[t] of y£ @tD + r [email protected] = f @tD with [email protected] = starter is given by t [email protected] = E-r t starter + E-r t Ÿ0 Er s f @sD „ s. Explain this: If r < 0 then, unless starter = 0, the plot of y[t] will NOT eventually merge with the plot of t [email protected] = E-r t Ÿ0 Er s f @sD „ s. · L.11) Here are three plots of solutions of a certain forced exponential diffeq: 150 100 50 5 10 15 20 t - 50 You are given that all three plots are all either solutions of a) y£ @tD + 0.15 [email protected] = [email protected] tD + 2 [email protected] tD or solutions of b) y£ @tD - 0.15 [email protected] = [email protected] tD + 2 [email protected] tD. Make your choice and say how you arrived at it. · L.12) Here are plots of the solutions of diffeq a): y' @tD + 0.01 [email protected] = 0 diffeq b): y£ @tD + 0.3 [email protected] = 0, diffeq c): y£ @tD + 1.3 [email protected] = 0, diffeq d): y' @tD + 2.3 [email protected] = 0, all with the same starter [email protected] = 7. The plots are not in order. Your job is to match the differential equation with the plot of its solution. Plot A Plot B 7 6 5 4 3 2 1 t 0 2 4 6 8 1012 7 6 5 4 3 2 1 t 0 2 4 6 8 1012 Plot C 7 6 5 4 3 2 1 t 0 2 4 6 8 1012 diffeq a) ------> Plot....... diffeq c) ------> Plot....... Plot D 7 6 5 4 3 2 1 t 0 2 4 6 8 1012 diffeq b) ------> Plot....... diffeq d) ------> Plot....... diffeq a) ------> Plot....... diffeq c) ------> Plot....... diffeq b) ------> Plot....... diffeq d) ------> Plot....... · L.13) Do you expect solutions of y£ @tD - 0.6 [email protected] = 0 to decay to 0 as t gets large? Why or why not? · L.14) Here are plots of solutions of the forced exponential diffeq y£ @tD + 0.8 [email protected] = f @tD with [email protected] = 4.0 for four choices of forcing functions f[t]: Plot A 5 -2 -4 70 60 50 40 30 Plot B 6 4 2 10 15 20 t 6 4 2 20 10 5 -2 -4 Plot C 10 15 20 t Plot D 6 4 2 · L.18) Here are plots of solutions of y£ @tD = 0.6 [email protected] with [email protected] = 2 and [email protected] y£ @tD = 0.6 [email protected] I1 - 100 M with [email protected] = 2 : 6 4 2 t t 5 10 15 20 5 10 15 20 -2 -2 -4 -4 The four forcing functions f[t] used in these plots are: [email protected] = 0.2 t, : [email protected] = 5 [email protected] - 6D, -0.1 t [email protected] = 2.3 E [email protected], [email protected] = 5 [email protected] tD. Your job is to match the plots to the forcing functions: f1[t]--------------> Plot..... f2[t]--------------> Plot..... f3[t]--------------> Plot..... f4[t]--------------> Plot..... · L.15) Give the formula for the solution of y£ @xD = 0.2 [email protected] with [email protected] = 1 and pencil in a rough sketch of the solution on the axes below. t 1 2 3 4 5 6 Explain why as t advances from 0, the plots of both of these solutions had no choice but to share a lot of ink initially. Explain why for larger t's, the plots had no choice but to pull apart, with one plot eventually sailing way above the other. · L.19) Here are six plots: They are plots of: a. a solution of y£ @xD = r [email protected] with r > 0. b. a solution of y£ @xD = r [email protected] with r < 0. c. none of the above Which is which? The plotted point is on the curve. [email protected] 60 50 40 30 20 Plot 1 y 14 12 10 0 2 4 6 810 12 Plot 2 x [email protected] 6 4 2 x - 2 10 20 30 40 Plot 3 8 6 4 2 [email protected] 8 6 4 Plot 5 2 4 6 8 x 10 12 · L.16) Give the formula for the solution of y£ @xD = - 0.5 [email protected] with [email protected] = 6 and pencil in a rough sketch of the solution on the axes below. The plotted point is on the curve. y 8 [email protected] 6000 5000 4000 3000 2000 5 10 15 20 [email protected] 8 6 4 Plot 4 10 20 30 40 [email protected] 12 10 8 6 4 4 2 2 4 6 8 x · L.17) You are given that [email protected] = 37.3 E-0.045 t Write down a differential equation that [email protected] solves. Include starter data. x Plot 6 x x 10 20 30 40 10 30 50 20 40 60 Plot 1 ---->............ Plot 2 ---->............ Plot 3 ---->............ Plot 4 ---->............ Plot 5 ---->............ Plot 6 ---->............ · L.20) aL You are given a solution [email protected] of y£ @tD + 5 [email protected] = 6 [email protected] - 4.7D What happens to [email protected] at t = 4.7 ? bL How does your response indicate that [email protected] - 4.7D ! 2 [email protected] - 4.7D ? 6 x · L.21) Here is the part of the plot of the solution of y£ @tD + 1.1 [email protected] = 5 [email protected] - 3D with [email protected] = 5. Your job is to sketch in the rest of the plot. [email protected] 5 4 3 2 1 0 0 2 · L.22) a) Write down the values of Ÿ0 4.99999 Ÿ0 5.00001 4 6 8 t [email protected] - 5D „ t [email protected] - 5D „ t Ÿ5.00001 [email protected] - 5D „ t. 100 Et [email protected] - 3D „ t b) Write down the values of Ÿ0 2.99999 3.000001 Ÿ2.999999 100 000 Ÿ3.00001 Et [email protected] - 3D „ t Et [email protected] - 3D „ t c) Here's a plot of UnitStep[t - 2]: 1.0 0.8 0.6 0.4 0.2 1 2 3 And here's a plot of UnitStep[t - 4]: 4 5 6 t 1.0 0.8 0.6 0.4 0.2 t 1 2 3 4 5 6 Digest the two plots and explain the result of this Mathematica calculation: [email protected], xD; AssumingBt œ Reals, ‡ Sin@xD DiracDelta@x - 5D „ xF t HeavisideTheta@- 5 + tD Sin@5D 0 ...
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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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