S251ex1(fa02) - Math 251 Fall 2002 Midterm Exam 1 1(i Linear homogeneous oder 2(ii Nonlinear order 1(iii Linear nonhomogeneous order 3 2(a y 00 4 y

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Unformatted text preview: Math 251 Fall 2002 Midterm Exam 1 1. (i) Linear, homogeneous, oder 2. (ii) Nonlinear, order 1. (iii) Linear, nonhomogeneous, order 3. 2. (a) y 00- 4 y + 13 y = 0 λ 2- 4 λ + 13 = 0 λ 1 , 2 = 2 ± 3 i y ( t ) = C 1 e 2 t cos 3 t + C 2 e 2 t sin 3 t (b) y 00 + 2 y + y = 0 λ 2 + 2 λ + 1 = 0 λ 1 , 2 =- 1 y ( t ) = C 1 e- t + C 2 te- t 3. (a) y = cos t + 1 y , y (0) =- 3 ydy = (cos t + 1) dt y 2 2 = sin t + t + const (- 3) 2 2 = sin 0 + 0 + const const = 9 2 y ( t ) = + s 2 µ sin t + t + 9 2 ¶ (b) ty + 3 y = 3 , y (1) = 2 y + 3 t y = 3 t μ ( t ) e R 3 t dt = e 3 R = t 3 y ( t ) = R t 3 3 t dt t 3 = t 3 + const t 3 = 1 + const t 3 2 = 1 + const 1 3 const = 1 y ( t ) = 1 + 1 t 3 1 4. (a) D (0) = 1000 , dD dt = 1 10 D ( t )- 120 (b) D ( t ) = 120 1 10 + Ce 1 10 t 1000 = 1200 + Ce C =- 200 D ( t ) = 1200- 200 e 1 10 t (c) D ( t ) = 0 1200- 200 e 1 10 t = 0 e 1 10 t = 6 t = 10 ln 6 5. (a) Both ln | t- 10 | and 1 t 2- t must be continuous, that is t- 10 6 = 0 and t 2- 2 6 = 0 and therefore we have 3 ”holes” in the real line: 0, 1, and 10. The maximal interval withouttherefore we have 3 ”holes” in the real line: 0, 1, and 10....
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This note was uploaded on 07/23/2008 for the course MATH 251 taught by Professor Chezhongyuan during the Spring '08 term at Pennsylvania State University, University Park.

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S251ex1(fa02) - Math 251 Fall 2002 Midterm Exam 1 1(i Linear homogeneous oder 2(ii Nonlinear order 1(iii Linear nonhomogeneous order 3 2(a y 00 4 y

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