Diff Eq Exam 3 (1994)

# Diff Eq Exam 3 (1994) - 110.302 Differential Equations Exam...

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Unformatted text preview: 110.302 Differential Equations Exam 3 April 19, 1994 Some formulas: 7r 1 7r 7r a0=lf max, an=—/ newsman, bn=1/ f<x>sinnxdw 71' ~71" 7T —7T 7T —'7T f1 952 961 f1 I f2 92 I 1/1 f2 : ———— ’U 2 U1 W ’ 2 W No books, notes, or calculators are permitted. Express all answers using real numbers and real functions only. To receive full credit you must Show all of your work and circle your answers. 1. (20 points) Find the Fourier series of the function: —1, —-7T < CE < 0 17 0<93<7r You should give your answer in the form f<x>=l 1+ + + +--» where represents a non—zero term of the Fourier series. 2. (30 points) In your answers for this problem, you may use n to represent an arbitrary positive integer. a) Find all the values of A so that the differential equation u” + Au 2 0 has a solution (other than u E 0) with boundary conditions u(0) = u(2) = 0. b) Find all the solutions of the partial differential equation 82y 82y 25 a? — a of the form yt’h t) = “(1?)7105) with boundary conditions u(0) = u(2) = 0 and initial condition 613(0) — 0 dt — 0 continued on reverse 0 3. (20 points) In this problem, you do not need to give solutions to the systems of differ— ential equations. a) Describe the critical point (0,0) of the system {jg—f = 3x—4y g— : 2x—3y using all the applicable words from the Word List below. b) Describe the critical point (0,0) of the system d—y = 5x+3y d9: _ dt using all the applicable words from the Word List below. 4. (30 points) Consider the linear system: {fl—f = .r—3y 2—3: = 3x—5y a) Find the general solution of the system. b) Describe the critical point (0,0) of the system using all the applicable words from the Word List below. Word List: asymptotically borderline center node saddle—point spiral stable unstable ...
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Diff Eq Exam 3 (1994) - 110.302 Differential Equations Exam...

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