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# Ord Dif Eq MATH 331

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This 8 page Class Notes was uploaded by Reuben Hudson DDS on Friday October 30, 2015. The Class Notes belongs to MATH 331 at University of Massachusetts taught by Staff in Fall. Since its upload, it has received 21 views. For similar materials see /class/232227/math-331-university-of-massachusetts in Mathematics (M) at University of Massachusetts.

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Date Created: 10/30/15

Math 3311 Review problems Exercise 1 Review exercise 19 chapter 3 p 375 Exercise 2 Review exercise 21 chapter 3 p 375 Exercise 3 Review exercise 23 chapter 4 p 445 Exercise 4 Review exercise 11 chapter 6 p 622 dY Exercise 5 Consider the linear systems E AY where A is given by K712 31gt blti 31gt Clti 1 dlt71171i4gt 5 31gt In each case of the ve cases 1 Determine the type of the system ie sink source saddle center spiral source spiral sink center degenerate eigenvalues 2 Draw the phase portrait of the system If the eigenvalue are real you need to compute the eigenvectors and indicate them clearly on the phase portrait If the eigenvalues are complex you need to determine the orientations of the oscillations clockwise or counterclockwise OJ Draw a rough graph of a typical solution xt Note that you do not need to solve the system to do this If the eigenvalues are complex indicate clearly in your graph the period of the oscillations dY Exercise 6 Consider the linear systems E AY where A is given by mltg gtw ifigt wltjgt wltj7 wlti gt In each case dY 1 Compute the general solution of E AY dY 2 Solve the initial value problem W AY Y0 lt 1 dY Exercise 7 Consider the linear systems E AY where A is given by mltiigt wltffgt and a is a parameter Use the trace determinant plane to determine the different types of the systems and the bifurcations of the systems as the parameter a increases on the real line Exercise 8 Consider the second order equation dzy dy 47 k7 5 0 dt2 dt y where k is a parameter with 700 lt k lt 00 As k varies describe using the tracedeterminant plane the different types of the systems and the bifurcations Exercise 9 Consider the second order equation dzy dy 7 k7 2k 0 dt2 dt y where k is a parameter with 700 lt k lt 00 As k varies describe using the tracedeterminant plane the different types of the systems and the bifurcations Exercise 10 Compute the general solution for the dzy dy 4 a w t a 6y 5 dzy dy 2t 1 W a 7 6y 7 e 2 c 76y et572t d2 d d TISd7il76y t21 Exercise 11 Consider the equation d 2 d i 4 7y 6s1n3t 1 Find the general solution 2 Describe the behaVior of the general solution as t 7 00 and graph a typical solution 3 Compute the amplitude and phase angle Exercise 12 Solve the initial value problem 01y2 dy i W 7 4 7 5y 6s1n3t7 y0 230 71 Exercise 13 Consider the equation d 2 712 8y 6sin3t 1 Determine the frequency of the beating 2 Determine the frequency of the rapid oscillations 3 Give a rough sketch of typical solution indicating clearly the results obtained in 1 and 2 Remark To answer this questions you do not need to compute the solutions explicitly Exercise 14 Find the general solution of d 2 a 712 16y 3sin4t d 2 b 712 16y 5cos2t Exercise 15 Solve the initial value problem d 2 i a 732 My 3mm ylt0gt1y lt0gt 0 d 2 b 712 16y 5cos2t y0 240 2 Exercise 16 Compute the inverse Laplace transform of the following functions 7 5785 5 55 1 5729 as2 bs2 C 322372 d 32232 e 32232 237 5 5 39 5 29 77 ll 9 322s2 321324 371324s5 Exercise 17 The function ht is given by 0 if 0 g t lt1 1125 2 if 1 tlt3 0 if 3 t 1 Compute the Laplace transform of ht Hint Write h as a combination of uat for suitable a s d 2 Solve the equation dig 1 3y ht Exercise 18 Use the Laplace transform method to solve the following initial value problems 1 1 dig 1 5y 574205 y0 77 Make also a graph of the solutions 2 i 4y 73u4t52t 47 y0 2 What is 11min ya d 2 3 772 4y 27120 cos3t72 y0 0 we 1 div2 7071 A 4 W 1 4y 3u1te y0 07 y 0 1 How does the solution behave for large 25 U a 7 74 dyz dy 7 27 10 t 1th 125 y U4 solution Li 12 d22 dyz y0 2 yO 0 What is limtH00 yt7 Make a graph of the 5y 65t y0 2 yO 1 Make a graph of the solution dtz 7y 65t y0 6 yO 71 Make a graph of the solution Math 3311 Review problems dY Exercise 1 Consider the linear systems E AY where A is given by M32 31 at 31 ltcgt 5 ltdgt171 4gt ltegti 31 In each case of the ve cases 1 Determine the type of the system ie sink source saddle center spiral source spiral sink center degenerate eigenvalues to Draw the phase portrait of the system If the eigenvalue are real you need to compute the eigenvectors and indicate them clearly on the phase portrait If the eigenvalues are complex you need to determine the orientations of the oscillations clockwise or counterclockwise 03 Draw a rough graph of a typical solution zt Note that you do not need to solve the system to do this If the eigenvalues are complex indicate clearly in your graph the period of the oscillations dY Exercise 2 Consider the linear systems E AY where A is given by K712 31gt blti 31gt 33 1 dlt7117 1 5 1 In each case dY 1 Compute the general solution of E AY dY 2 Solve the initial value problem E AY Y0 lt 1 dY Exercise 3 Consider the linear systems E AY where A is given by ltagt 1 ltbgtlti 34 and a is a parameter Use the trace determinant plane to determine the different types of the systems and the bifurcations of the systems as the parameter a increases on the real line Exercise 4 Consider the second order equation dzy dy 47 k7 5 0 it dt y where k is a parameter with foo lt k lt 00 As k varies describe using the tracedeterminant plane the different types of the systems and the bifurcations Exercise 5 Compute the general solution for the b 76y52t C 76yiete d 76yt21 Exercise 6 Consider the equation dyz dy W 4 1 7y 6s1n3t 1 Find the general solution 2 Describe the behavior of the general solution as t a 00 and graph a typical solution 3 Compute the amplitude and phase angle Exercise 7 Solve the initial value problem dyz dy i W 74 7 5y 6s1n3t7 y0 23 0 71 Exercise 8 Consider the equation dyz it 1 Determine the frequency of the beating 1 8y 6sin3t 2 Determine the frequency of the rapid oscillations 3 Give a rough sketch of typical solution indicating clearly the results obtained in 1 and 2 Remark To answer this questions you do not need to compute the solutions explicitly Exercise 9 Find the general solution of 2 a di16y 3sin4t dt2 d 2 b i16y 5cos 2t dtz Exercise 10 Solve the initial value problem d 2 i a 712 16y 3s1n4t y01y0 0 d 2 1 732 16y seesaw yltogt 2y lt0gt 2 Exercise 11 Compute the inverse Laplace transform of the following functions 7 678s 6759 1 6729 as2 bs2 C 322372 d 32232 e 32232 23 7 5 5 39 5 29 h l 9 32232 321324 37132435 Exercise 12 The function ht is given by 0 if 0 g t lt 1 1125 2 if 1 tlt3 0 if 3 g t 1 Compute the Laplace transform of ht Hint Write h as a combination of uat for suitable a s i dy 2 Solve the equation a By ht Exercise 13 Use the Laplace transform method to solve the following initial value problems 1 139 dig 5y 5mg7 340 77 Make also a graph of the solutions 1 2 7 4y 3u4t52t747 y0 2 What is th W dyz dt2 d 2 439 T 4y 3u1teit71 M0 07 y 0 1 How does the solution behave for large 15 3 1 4y 2u2t cos3t 7 2 y0 O7 yO 1 2 5 W 2 10y u4t y0 27 yO 0 What is limtH00 yt7 Make a graph of the solution 6 1 5y 65t y0 27 yO 1 Make a graph of the solution 7 1 4 7y 65t y0 67 yO 71 Make a graph of the solution Exercise 14 For the following problems 1 Compute the Laplace transform of the solution 2 Find the poles of the Laplace transform of the solution 3 Discuss the behavior of solution make a schematic graph showing clearly the behavior for large 25 Remark To answer these questions you do not need to compute the inverse Laplace transfrom7 ie no partial fraction expansions are necessary a di 4 4 104 7 uilttgtcoslt3ltt e 4 dtZ dt d 2 d b 172 4 4 4 10y 7 u4t52ltt4gt cos3t 7 4 d 2 d c 772 4 4 410 7 u4t52ltt4gt cos3t 7 4 d 2 d 732 4 3 710 7 u4t cos3t 7 4 d 2 d dsz2 4 3 710 7 u4te2ltt4gt cos3t 7 4

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