Review Sheet for MATH 331 at UMass
Review Sheet for MATH 331 at UMass
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Date Created: 02/06/15
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 3 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 4 ka5y 0 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 9 Consider the second order equation Q d dt2kdii2ky0 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 10 Compute the general solution for the b 76 5 c 76y5t572 d 76yt21 Exercise 11 Consider the equation 01y2 dy i W 4 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 12 Solve the initial value problem 01y2 dy i W 7 4 7 5y 6s1n3t7 y0 230 71 Exercise 13 Consider the equation dyz dtz By 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 d2 a y 16y3sin4t dt2 2 b di16y 5cos 2t 1t Exercise 15 Solve the initial value problem d 2 i a 732 My 3mm ylt0gt1y lt0gt 0 dyz b 7752 16y 5cos2t y0 2y 0 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 23 7 5 5 39 5 29 h 9 322s2 321324 Z371824s5 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 1 139 dig 5y 5mg7 340 77 Make also a graph of the solutions 2 i 4y 7 73u4t52t 47 y0 2 What is 11min ya dyz 3 dtz 1 4y 2u2t cos3t 7 2 y0 07 y 0 1 d 2 4 d 1 4y 3u1te t 1 y0 O7 yO 1 How does the solution behave for large 25 2 5 31 2 109 U4t 240 2 34 0 0 What is limtH00 yt Make a graph of the solution 1342 6 dtz l 5y 65t 340 2 40 1 Make a graph of the solution 1342 dy 739 w l 4E l 7y 65t 340 6 34 0 71 Make a graph of the solution
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