MTH 205-Solutions of Review Problems-Summer 2009 - Copy

MTH 205-Solutions of Review Problems-Summer 2009 - Copy - V...

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Unformatted text preview: V Math 205, Dz'ﬁerential Equations 6) Review Questions for F ﬁnal Exam 1. (1 points). Use Laplace T ransform to solve the following system of dif- ferential equations (U=Unit step function, 6=Dirac delta function): :L"(t) + y(t) = Ll(t — 2) :U(t) + y’(t) = 6(t — 3) such that ;L‘(O) = O and y(0) = O. .2. (Iffomk) Fwd +l~e We,» (a) ¢ffmd (at) ifﬁtiljm: W) %ii*\$‘“us (V) A YewhuAfmrolti " than La’o/‘can WM\$§0t'm 3'2 ) osl; (z -’ d If? Vase “Ll/[ace "f 5' } a-‘mz’ﬁp—Wi} (C) Sf— gi/og-rzli {(+57 Ll‘ (20ft);th grove. ’“ne ;O\\DQ‘A<AV in.f{J,Va(uaP./alal6n£- 0L “Vt/+3?) v- 2179-96—1 3(1) =- /20 7, 7, i \o‘ V :33 .-7.:S ) Véfkcﬂ , 1L \gJM‘ lg n. Bulb/ugh :f’“” ,9 MM, .1 (‘OPa-inliE De, CXMch, U DO “Wk CVA‘M‘LC‘ Method 095 UAAﬁ‘l‘e‘“M¢A M‘%9S‘°‘M ' Conj‘l'ﬂn‘l’k ’ (2x\ on U :34,sz at} +3} +196 1“ «'3; 5’ “89’3" ’gu/Hz 6. (15 points) Answer only one of the following two application questions. eapot and cools. The temperature tea has a temperature of 191°F. the water is added. A. Boiling water (212°F) is poured into a t of the room is 65oF. After 2 minutes the f the tea be 16 minutes after .. B. A mass weighing 4 lb stretches a spring 2 ft. The mass is attached to a viscous damper with a damping force numerically equals to the instantaneous velocity. If the mass is set'in motion 1 ft above the equilibrium position with a downward velocity of 8 ft. / sec, find (a) Its position :1: at any time t. (b) Determine when the mass ﬁrst return to its equilibrium position. 7. (’15 ﬂat-rite) Answer True (T) or False (F). a. _.___~____ The first step in solving 3/” + ély’ + y = F(;1:) is to solve y” + 421/ + y = 0- b. ______ The frequency for at” + 4x = 0 is 4. c. _____ All the solutions of the differential equation 3/” + 4y = cos 2:1: can be found by substituting y = A cos 2.1: + Bsin 2:73 into the equation and solving for A and B. d. ________ The steady-state solution for x” + 4x’ + 41: = 8 is a: = 2. e. _.__. The integrating factor for the following differential equation emy’ = y — 1 is 6"”. ' n f. The general solution of the differential equation :rrgy” — 3:7:y’ + 43/ = 0, 93 > O is 1 = Ag:2 + 313 lnm. [I I i. i , 4“ g. _____ The differentfél’equation (5y + 2.7?)y’ = 22/ is ‘eka‘dt’. h. ___ The equation 271/ = yex/y — a: is classiﬁed a homogeneous differential equation. i. _..___________ The two functions f1 = a: and f2(:r) 2 lat] are linearly dependent on the interval (—00, 00). j. __ The differential equation xy’ = y(:vy3 — l) is Bernoulli, and can be solved by using the substitution 'u. = y‘2. 8. (10 points) Use the power series method to solve the given differential equation. Determine the recurrence relation and the first seven terms. y” + .L‘y' + 112,1; = 3 + :r, y(0) = 3‘ y’(()) = _5 9. (10 points) Solve the following Initial Value Problem using undetermined coefficients: 1/” — 103/ + 251/ = 50:1: + 685'”, y(0) = I ’(0) — So‘u-HM: 1) er—Qmmw I M -- zes- x +3 :LLHL-t) I 2) ’ﬂfeuasa‘t}: {fast} 1 SHS-Z 1 +3’ : SUN-3)) manager“) 51‘ £fﬂ+£§u3=ac§wcm§ — f , \ 5X6) Xto ‘5 s +?\$ 5H5”. 3‘83 + K‘s) .— S; : 54 \$2 chs) +\((s, : e’! (5-3)“...15' 1 - .. ‘ 7. 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MTH 205-Solutions of Review Problems-Summer 2009 - Copy - V...

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