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Unformatted text preview: MAT 485 Final Exam 13 Dec 2004 Prof. V. Fatica Name ,
W SUID W l. Solve the initial value problem: 2. Use an augmented matrix and GaussJordan elimination (RREF) to find the solutions to
the following system oflinear equations. x+y+z=4
x+2y—z=7
2x+5y~4z=17 y—22=3 3. Determine whether the following matrix in invertible, and ifit is invertible, give its
inverse. l 5 3
l 3 —l
0 l 2 Does the following set of vectors form a basis of R3? Justify your eonclusion. (Hint: use
the result above.) 4. Consider the damped. forced harmonic oscillator governed by the differential equation + 2y’ + y = 4cost
(a) Find the general solution, yh , ofthe related homogeneous equation
,V"+2y'+y= 0.
(b) Find a particular solution, yﬂ , ofthe non—homogeneous equation. [The method of undetermined coefficients will work well] (c) Give the general solution of the nonhomogeneous system.
(d) Describe the longterm (I —> oo) behavior ofthe oscillator. x' = 2.x +
5. Consider the following system ofdifferential equations: I y . After
y = 2x + 3y computing the necessary “eigenstuff", (a) give the general solution ofthe system, . 0 3
(b) ﬁnd the particular solution which satisﬁes q =[ J, and
y(0) 0 (c) sketch the xy phase—plane, showing the eigenvectors, the particular solution found
in part (b), indicating the direction of increasing I and the relation of that solution
to the eigenvectors. x'=xy+l 2 } has exactly one The (nonlinear) system ofdifferential equations { I
y = x + y equilibrium solution. Find the equilibrium solution and use the linearization ofthe
system at the equilibrium solution to classify the behavior there (as stable, unstable, attracting, repelling, node, saddle, spiral, center, et cetera). Find a formula for x(l) using slepO and ﬁnd the Laplace transform, X(s) ,of x(t). 8. Use Laplace transforms to solve the initial value problem: x'—x : step(1 —2), x(0) =1 ...
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This note was uploaded on 01/15/2012 for the course MAT 485 taught by Professor Staff during the Fall '11 term at Syracuse.
 Fall '11
 Staff

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