5389-Ans-Part-III - Part III Differential Equations...

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Unformatted text preview: Part III. Differential Equations Chapter 1 Exercises 1.1 3. 4. 7. 8. 10. 11. 14. 15. 17. 19. 21. y1 is a solution, y2 is not. Both y and z are solutions. r = −4, r = 2 r=3 r = 1, r = 2 r=2 (b) y = x2 ln x (c) and (d) No solutions; ln 0 is not defined. (a) y = C1 sin x for any constant C1 . (b) The initial conditions imply C1 = C2 = 0. xy − 2y + 6 = 0 y + y − 2y = 0 x2y − 2xy + 2y = 0 Chapter 2 Exercises 2.1 2. 3. 5. 7. 8. 9. y = x + Cx2 y= sin x C +2 x2 x y = x(ln x)2 + Cx y = x2ex − ex2 y= 2 Cx − 3x3 y1/2 = Ce2x − ex or y = (Ce2x − ex )2 1 Exercises 2.2 2. 3. 5. y = − ln (ex − xex + C ) 2 y−1 = Cex −2x y+1 y2 = 2 ln (ln Cx) − 1 e [1 + ex ] 1+e 6. y = ln ln 7. 9. 10. y + ln y = 1 x3 − x − 5 3 y= 1 Cx + 1 + ln x y = Cxey/x − xey/x ln x − x Exercises 2.3 1. 2. x2 + 3y2 = C (x − 1)2 + (y − 2)2 = C 2 t 4. (a) A(t) = 50e 2 ln (9/10) = 50 9 10 t/2 (b) 40.5 (c) 13.16 hours 6. 7. 8. (a) P (t) = 4.5e0.02310 t 132 years; 2112 8:52 pm (b) 48.19 years (c) 6.17 billion (approx.) Chapter 3 Exercises 3.2 2. 3. 4. Yes, their Wronskian is non-zero. (a) r = −1, 4 (a) r = −1, 4 (d) y ≡ 0 (b) {e−x , e4x }; y = C1e−x + C2e4x (b) {x−1, x4}; y = C1x−1 + C2x4 (c) y = 4 e−x + 1 e4x 5 5 (c) y = 9 x−1 + 1 x4 5 5 2 5. (a) y = x + 1 and y = x2 + 1 are solutions and they are independent. (b) y = C1 (x + 1) + C2(x2 + 1) 7. 9. 10. {x, x2} αδ − βγ = 0 Set u = y2 and calculate u . W ≡ 0 ⇒ u = 0 ⇒ u = C constant. y1 Exercises 3.3 1. 2. 5. 7. 9. 10. 11. y = C1e6x + C2e7x y = C1e5x + C2xe5x √ √ y = C1e−x cos x 2 + C2e−x sin x 2 y = −ex cos x y − 6y + 9y = 0 y + 2y + 10y = 0 (a) y = (b) y = 12. 15. 16. α + 2 2x 2α − 2 −x e+ e ; α = −2 3 3 β + 2 2x 4 − β −x e+ e ; β = −2 3 3 (b) α > 1 (a) ga < 0 y = C1x4 + C2 x−2 y = C1 x2 + C2 x2 ln x 3 ...
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This note was uploaded on 01/06/2011 for the course MATH 3333 taught by Professor Staff during the Spring '08 term at University of Houston.

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5389-Ans-Part-III - Part III Differential Equations...

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