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# Engineer Math IV Differential Equations 22M 034

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This 7 page Class Notes was uploaded by Baron Will on Friday October 23, 2015. The Class Notes belongs to 22M 034 at University of Iowa taught by Staff in Fall. Since its upload, it has received 41 views. For similar materials see /class/227996/22m-034-university-of-iowa in Mathematics (M) at University of Iowa.

## Reviews for Engineer Math IV Differential Equations

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Date Created: 10/23/15

Review for EXAM III22M034 The third exam is on Wed 1119 it is a 50 minute m class exam How to prepare for the exam The best way to prepare for the exam is to make sure that you understand and know how to use all the concepts theorems formulas problems etc from your notes Go over the homework problemsido them again DO NOT rely exclusively on the review problems Material covered Sec 636465717273 Allowed to use during the exam One single letter sized sheet of paper 85 gtlt 11 with hand written notes of your choice on front and back You will also be provided with a sheet containing elementary Laplace transforms see Table 621 on p 319 NO calculators NO books and no other aids will be allowed H 3 Review Monday 1117 in class Student ID BRING YOUR ID FOR THE EXAM Review problems Find the Laplace transform of N t 5ust t 1U2t Solution ln order to apply the general formulas from 63 about Laplace transform of products of functions involving step functions we rst write ft as N t 3u3t t 2u2t 21130 u20 Consequently we get that 6 35 6 25 17 2se 35 7 1 se 2S ft 5 39 t 7 5 29 t 7 2 s s 52 Find the inverse Laplace transform of the following functions a H 8 b Gs figs dont use partial functions Solution 9 7 5 a We have that Hs 6729923572 and hence 1Hs u2tft 7 2 where f ilfm 6t 7 6 2 Consequently 1Hltsgt gum 7 57w b 271mg gy cost Use the Laplace transform to solve the following initial value problem 5 u u Eu ht u0 uO 0 where Mt sint if0 tlt7r 0 if t 2 7139 Solution To nd the Laplace transform of the function ht we write it as ht sint uwt sint 7 7T By taking the Laplace transform of the equation in question we get that 1 1 Y s 6 lt521gtlt52s3gt lt521gtlt52s3gt l 3 where Ys Note that 1W125 74costs1nt4e 2 cost 6 5 sint of course to nd this inverse transform one needs can use partial fractions and then the table with the basic Laplace transforms Denote this function ft Putting everything together we get that the solution to the above initial value prob lem is W VIC5 ft uiff1t 7T Use the Laplace transform to solve the following initial value problems a y 7 y 5t 7 4W 00307 110 MO MO y 0 0 b y y 6t 7 27139 cost y0 0y 0 1 Solution Did it in class a Taking the Laplace transform ofthe equation above we get that Ys 47rs 4171 807 ya U4 t smht747r275mt747r b yt sint uh sint 7 27139 6 Verify that the given vector 6 0 Xt 78 e 2 1 62 74 71 03 5 satis es the differential equation 1 1 x 2 71 x 0 71 1 Solution This is just a direct veri cation Determine whether 2 0 7 Km 1 x2 1 X3 2 0 0 0 are linearly independent Solution They are linearly dependent Just look at the matrix determined by these three vectors and compute its determinant Indeed this matrix which has columns x123 has the last row consisting of zeros Therefore its determinant is indeed zero Find the eigenvalues and eigenvectors of the matrix 3 2 2 A 1 4 1 72 74 71 Solution To nd the eigenvalues we need to solve the characteristic equation p detA 7 AI 0 for A Now it is not dif cult to see that the characteristic polynomial of A is p A 7 1 7 2 7 3 Consequently the eigenvalues are A1 1 A2 2 and A3 3 To nd the eigenvectors and the eigenspace corresponding to A1 1 we want to nd 1 2 3 all vectors x such that Ax Alx X When solving this system of equations for 12 zg we obtain that 2 0 and 1x3 0 So if we denote 1 c then Consequently7 xm 0 is the distinguished eigenvector for A1 meaning that 71 the eigenspace corresponding to A1 is spanned by xm For A2 27 the corresponding eigenspace is generated by one eigenvector zm 2510 For A3 37 the corresponding eigenspace is generated by one eigenvector z3 0151 Page 4 Review for EXAM II22M034 The second exam is on Wed 1029 it is a 50 minute m class exam How to prepare for the exam The best way to prepare for the exam is to make sure that you understand and know how to use all the concepts theorems formulas problems etc from your notes Go over the homework problemsido them again DO NOT rely exclusively on the review problems Material covered Sec 32 33 363738396162 Allowed to use during the exam One single letter sized sheet of paper 85 gtlt 11 with hand written notes of your choice on front and back You will also be provided with a sheet containing elementary Laplace transforms see Table 621 on p 319 NO calculators NO books and no other aids will be allowed 1 3 Review Monday 1027 in class Student ID Please bring your student ID for the exam Review problems a ls the given pair of functions linearly dependent or independent f1t 356117 f2t 560174 b Suppose that yl and y2 are linearly independent solutions of 1 7 y 7 3W aa 1y where a is a constant Assume that the Wronskian satis es Wy1y23 26 Find the Wronskian Wy1y2 t for all t gt 3 Solution 1 Note that f1 366 and f2t 54 f1t In other words f1 and f2 are linearly dependent b Rewrite the equation in its standard form with pt 73122 Using Abel7s theo rem we have Wy1y2t 054W Ctz 71 Since Wy1y2 3 26 we obtain that c 268 and so Wy1y2 268gt271fg A mass of 10lb stretches a spring 6271 The mass is displaced 10m in the positive direction from its equilibrium position and released with an initial velocity of 15 9 7 Cf 03 a Assuming that there is no damping7 and no external force acting on the mass7 set up the initial value problem describing the motion of the mass Do NOT solve the differential equation Taking the same spring mass system as in part a7 assume now that there is damping as follows The medium in which the mass moves exerts a viscous resis tance of 7lb when the the velocity of the mass is 5 Further7 the mass is acted upon by an external force of 3cos4tlb Set up the initial value problem of the motion of the mass under these additional circumstances Solve the differential equation 0 Solution Did in class Suppose the motion u of a mass hanging at the end of a spring satis es the initial value problem 1 Eu 32u 0u0 410 27 where the time t is measured in seconds and ut is measured in feet Find the position u of the mass at time t Determine the frequency7 the amplitude and period of the motion Solution To nd ut7 use characteristic equations7 etc This way we get that ut 4cos8t isin8t Consequently7 the amplitude is R 142 2 the frequency is 7r we 87 and nally the period is T ii Z Use the method of undetermined coef cients to nd the general solution of the non homogeneous differential equation 2 y i 21 t 2 Solution The general solution is yt 016 026 2 306 Consider the non homogeneous differential equation y 710y 34y 4 5 t i 3et sin3tt71265t cos3t Determine a suitable form for a particular solution Yt if the method of undetermined coef cients is to be used Do not determine the coef cients Solution Did in class Verify that y1t 6t and y2t t satisfy the corresponding homogeneous equation of 1 i ty W 7y 2t7125 t0 lt t lt 1 Find a particular solution of the equation above Page 2 5 00 Solution We did the rst part in class For the second part7 you may want to use the method of variations of parameters We deduce that a particular solution is Yt 72t71e t Determine the inverse Laplace transform of the following functions a 21 390 Solution The idea is to write Ys as a sum of simpler functions whose inverse trans forms are well known a The inverse transform is yt e Zt cost 7 106 sint b The inverse transform is yt 65 c0s5t 7 7i 5 sin t Use the Laplace transform to solve the following initial value problem 3y 7 4y 3y 62590 0710 1 Solution yt 162 7 e cos t 7 6 sin Page 3

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