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following differential equations, give the form of the â€œguessâ€
yo for the particular solution as explicitly as possible Without
actually ï¬nding the coefï¬cients. Thus, for example, for the
equation yâ€ + y = 62â€™3 my answer would be yo =2 A6â€. As an
aid the roots 7" for the characteristic polynomial are given. (a)
yâ€ + yâ€™ â€” 6y = estcos(2t), r = 2, â€”3. i N Hâ€” 3+ ï¬‚ ,,,,,,,,
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y â€”4y+4yâ€”â€”te 72.2. ,/~" &Â§b:A{lâ‚¬L+â€”\â€”\BJâ‚¬QL% *QJIRQH/ 10 (3) Use the method of undetermined coefï¬cients (educated
guessing) to ï¬nd the general solution to the following nonâ€”
homogeneous equation. Other techniques will not receive credit. yâ€+6yâ€™+9y=1+2eâ€˜3t.
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y y 9t+97r, 7r<t with the initial conditions y(0) = O, yâ€™(0) = 2. Assume that y
and yâ€™ are continuous at t 2 7T. 10 pts 3â€œ My 9 ngmuo wanton \fauonjzwk mtAHB
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'9â€˜ 4 pts 4 pts 12 (1) (5) Consider the following differential equation. yâ€ + (1 ~ ï¬‚)?!  ï¬‚y = 0
(a) For which values of ,8 does this equation have at least one
solution y such that liming, y(t) = 1? (b) Find 3/ given that y is a solution to (1) such that y(0) = 2
and limp,00 y(t) = 1. iquÂ°*+rse*rA+\se~* 13 (6) We wish to solve the following equation using variation of paâ€”
rameters: (2) yâ€ + yâ€™  6y = 90$)
(a) The method of variation of parameters, as taught in class, requires the use of a system of two linear equations in two. unknowns uâ€™l and 11/2 of the form
auâ€™l + buâ€™2 = c
duâ€™1 + euâ€™2 = f In the context of the given data, a =?,b =?,c =?,d =
?, e =?, f 2? Be explicit! \j\ \Jr \C "e L: \jl\lâ€œl\c[gdâ€˜l\
\f +\3\~h\j :0 Tltfâ€˜hï¬lb
(r +53CV~D=O Y3'lxl ateâ€”H hiï¬eï¬ {:0
(l: 'EQ'H a: all tight) 6 pts 14 (b) Find U1 and U2. Your answers may be expressed in terms
3 pts of integrals. \ â€˜ 1* (*3 '
\4â€˜ 1% :â€œlÃ©elï¬‚ll (c) Express the general solution to (2) on page 13 in terms
1 pts 0f 3J1, @2,U1,U2 \0: â€œâ€˜9â€˜+\'\5Â«\Â§L+A â€˜rgyl 15 (7) In this problem, the units are all English (feet, pounds, g =
32 ft / 3e02, etc. ) A weight of 4 lb stretches a spring 33 feet. The
spring is acted on by an external force of sin wt lb. Assume
there is no friction and the spring is initially at rest. 10 pts (a) Find the spring constant k. F Mr
\Q=Â§ 2%:\h;â€˜/ (b) Find the mass m of the object. / â€˜\ W2EÂ§Cï¬llÂ§ QDl'Sl: (0) Give an inital value problem (differential equation and
initial conditions) that could be solved to ï¬nd the disâ€” placement from rest u(t) of the spring at time t. Do not
solve the initial value problem. \Kâ€˜r hm =m<miwk3 is + um: 83 miwtl (d) For What value of ad would resonance occur? Wotwm:\lÂ§x:m :tï¬/ l O D 16 (8) An object of mass 1g attached to a spring with spring constant
k is sliding along a straight line on a surface which exerts a
frictional force With coefï¬cient of friction of 7. It is subject to
an external driving force described by the function F (t) (a) Write a differential equation that could be solved to ï¬nd a
formula for the displacement u(t) from rest of the object.
Your answer Will involve k and *y. W\\;C\:~\Q\)\â€˜\GVC +Hâ€™Cl
u,â€œ thawÂ» 4M Nâ€˜WE â€œWM HQ (b) Suppose that for a speciï¬c choice of k, 7, and F(t) our
solution is described by u(t) = cos 3t + 6â€”37: cos(2t) â€”~ Zeâ€”Btsin(2t).
(i) What is the steady state solution? m0) H z// (ii) What is the value of 7? sewsâ€”wGï¬gâ€”EC \nâ€” â€J'K/cmmi imam â€˜7 t: â€˜3 h
l' l 3sâ€œ:Wâ€˜*<wâ€˜+ a m â€”3 a (iii) What is the value 99k? bu w/ /
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