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# MAT514-2000Fall - December 187 2000 MAT 514 Final Exam NAME...

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Unformatted text preview: December 187 2000 MAT 514, Final Exam NAME: Instructions: Write your answers and Show all your work on this test. There are 9 problems on 8 pages, for a total of 200 points. To receive credit, you must Show all details of your work. . . . . dy t2 + 1 1.(20 pomts) Find the general solut1on of the equatlon a = 2 . y 2.(20 points) Draw the phase line, ﬁnd and classify all the equilibrium points for the autonomous equation ii: = 5y(y2 — 9) . If y(t) is the solution to this equation with y(1()) = 1 ﬁnd thin y(t) . . . . d 3.(30 points) Consider the one-parameter family of differential equations _y 2 (1y2 * 2y + 1 , where the parameter a is any real number. Find all the bifurcation values of the parameter (1. Draw the bifurcation diagram. 4.(20 points) Solve the initial value problem t: , dY 5.(20 points) Find the general solution of the linear system E = ( 31 i > Y . Then ﬁnd the particular solution with Y(O) = (1, 2) . dY 6.(20 points) For the linear system — < 2 1 E _ —2 0 librium point at the origin. Sketch the phase plane, including the solution curve with initial condition Y(0) = (1,0) . > Y , find the eigenvalues and classify the equi- 7.(20 points) Find and classify all the equilibrium points of the autonomous system 8.(20 points) Find the solution to the following forced harmonic oscillator equation: 3/” +9y = 6sin3t , y(0) 2 0, y'(0) = 2. Describe the long term behavior of this solution and sketch its graph. 9.(30 points) Solve the initial value problem 3/" + 4y' + 8y 2 267(t) , y(0) = O, y'(0) =1. Describe the long term behavior of the solution. 6.6 The Qualltatlvc Theory of Laplace Transforms 557 Table 6. l Frequently Encountered Laplace Transforms Y(S) = 1U] l )‘(0 = :01 Y(s) = (s > a) y(t) = In Y(s) = n. (x > 0) I _a 571+! yo) = sinwt Y(s) = w y(t) = coswt Y(s) = 5 32 +02 :2 +a)2 - y(‘)="m5l“w‘ Y(S):(s—a;+a,2 Y(')="mC°5W HS): (5 —:)_2:w2 E 2_ 2 g y([) = I sina)! Y(S) = (52 if??? y(t) = tcoswt Y(5) : (:2 + :2)2 e—as y(!) = uaU) Y(:) = (s > 0) y(t) = 6N) Y(;) = e’“ Table 6.2 Rules for Laplace Transforms: Given functions y(t) and w(t) with ID]: HS) and \$[wl = W(:) and constants a and a ‘ Rule for Laplace Transform Rule for Inverse Laplace Transform M dy 1[;]= 513M ~ y(0) = 5Y0): ,v(0) rly + w] = £in + :6th = no + we) :c—‘IY + W] = r1m+ :r'IWJ = y(:) + wm flay] =a\$ly1=aY<s> \$'1[aY]=a£‘l[Y]=ay(l) ilua(r)y(! — 0)] = e‘“‘£[yl = e‘“Y(:) = V \$"[e'“‘Y] = ua(r)y(! - 0) flea’yw = m —a) «Wm: — an = emx-Im : e‘“y<:> M ...
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MAT514-2000Fall - December 187 2000 MAT 514 Final Exam NAME...

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