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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 oneparameter 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 = emxIm : e‘“y<:> M ...
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