Exam 1 Review Answers

Exam 1 Review Answers - 1. ( i ) y 1 = x 4 , y 2 = x-2 . (...

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ANSWERS Exam 1 Review Part I. y is/is not a solution. 1. Both are solutions. 2. Both are solutions. 3. y 1 is a solution, y 2 is not. 4. Both are solutions. 5. y is a solution. Part II. Find the differential equation. 1. 2 xyy ± =3 y 2 +6. 2. x 2 y ±± - xy ± - 3 y =0 . 3. y ±± +4 y ± +4 y =0 . 4. xy ±± - 2 y ± =0 . Part III. Identify. 1. Separable 2. Separable 3. Homogeneous 4. Bernoulli 5. Linear 6. Separable 7. Bernoulli Part IV. First order linear equations. 1. y = 1 2 x 2 sin 2 x + Cx 2 . 2. y = C - x 1+ x 2 . 3. y = x (ln x ) 2 + Cx . 4. y = e x x 2 - e x x 3 + 2 x
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Part V. Separable equations. 1. y 2 =1+ Ce - x 2 . 2. e y = 1 e x - xe x + C . 3. ln | y 2 - 1 | = x 2 - 2 x + C . 4. y = C ln x . 5. y +ln | y | = 1 3 x 3 - x - 5. Part VI. Applications. 1. 1. y 2 +2 x = C . 2. 2. y 3 =3 x + C . 3. ( i ) V ( t ) = 200(4 / 5) t/ 4 . ( ii ) V ( t ) = 200(4 / 5) t 2 / 16 . ( iii ) V ( t )= ±² 2 10 - 5 2 ³ t + 200 ´ 2 4. (a) A ( t ) = 50(9 / 10) t/ 2 . (b) A (4) = 50(9 / 10) 2 . (c) 13.16 hrs. 5. (a) 1 / 27 sq. cm. (b) T = 4ln2 ln 3 . 6. (a) 40 . 1 . (b) 1.62 min. 7. (a) dP dt = k ( M - P ). (b) P ( t )= M - M (7 / 10) t/ 10 . (c) t = 10 ln(1 / 10) ln(7 / 10) . Part VII. Second order linear equations.
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Unformatted text preview: 1. ( i ) y 1 = x 4 , y 2 = x-2 . ( ii ) y = C 1 x 4 + C 2 x-2 . 2. (a) r 1 = 4 , r 2 =-1. (b) y = C 1 x 4 + C 2 x-1 . (c) y = 1 5 x 4 + 9 5 x-1 . (d) y 0. 3. (a) y 1 = x + 1 , y 2 = x 2 + x + 2. (b) y = C 1 ( x + 1) + C 2 ( x 2 + x + 2). Part VIII. Homogeneous equations with constant coecients. 1. y = C 1 e-x + C 2 xe-x . 2. y = C 1 e 5 x + C 2 e 3 x . 3. y = e x ( C 1 cos 2 x + C 2 sin 2 x ). 4. y =-e x cos x . 5. y = 7 e-2( x +1) + 5 xe-2( x +1) . 6. y + 6 y + 13 y = 0. 7. y + (4- ) y -4 y = 0 , any real number. 8. y -8 y + 16 y = 0. 2...
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Exam 1 Review Answers - 1. ( i ) y 1 = x 4 , y 2 = x-2 . (...

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