Calculus: One and Several Variables

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-09 JWDD027-Salas-v1 November 25, 2006 19:21 SECTION 9.1 481 CHAPTER 9 SECTION 9.1 1. y 1 ( x ) = 1 2 e x/ 2 ; 2 y 1 y 1 = 2 ( 1 2 ) e x/ 2 e x/ 2 = 0; y 1 is a solution. y 2 ( x ) = 2 x + e x/ 2 ; 2 y 2 y 2 = 2 ( 2 x + e x/ 2 ) ( x 2 + 2 e x/ 2 ) = 4 x x 2 = 0; y 2 is not a solution. 2. y 1 + xy 1 = xe x 2 / 2 + xe x 2 / 2 = 0; not a solution y 2 + xy 2 = Cxe x 2 / 2 + x + Cxe x 2 / 2 = x ; y 2 is a solution. 3. y 1 ( x ) = e x ( e x + 1) 2 ; y 1 + y 1 = e x ( e x + 1) 2 + 1 e x + 1 = 1 ( e x + 1) 2 = y 2 1 ; y 1 is a solution. y 2 ( x ) = Ce x ( Ce x + 1) 2 ; y 2 + y 2 = Ce x ( Ce x + 1) 2 + 1 Ce x + 1 = 1 ( Ce x + 1) 2 = y 2 2 ; y 2 is a solution. 4. y 1 + 4 y 1 = 8sin2 x + 8sin2 x = 0; y 1 is a solution. y 2 + 4 y 2 = 2cos x + 8cos x = 6cos x ; not a solution. 5. y 1 ( x ) = 2 e 2 x , y 1 = 4 e 2 x ; y 1 4 y 1 = 4 e 2 x 4 e 2 x = 0; y 1 is a solution. y 2 ( x ) = 2 C cosh2 x, y 2 = 4 C sinh2 x ; y 2 4 y 2 = 4 C sinh2 x 4 C sinh2 x = 0; y 2 is a solution. 6. y 1 2 y 1 3 y 1 = e x + 18 e 3 x 2( e x + 6 e 3 x ) 3( e x + 2 e 3 x ) = 0; not a solution y 2 2 y 2 3 y 2 = 7 4 (6 + 9 x ) e 3 x 2(1 + 3 x ) e 3 x 3 xe 3 x = 7 e 3 x ; y 2 is a solution. 7. y 2 y = 1; H ( x ) = ( 2) dx = 2 x, integrating factor: e 2 x e 2 x y 2 e 2 x y = e 2 x d dx e 2 x y = e 2 x e 2 x y = 1 2 e 2 x + C y = 1 2 + Ce 2 x 8. y 2 x y = 1; H ( x ) = 2 x dx, integrating factor: x 2 x 2 y 2 x 3 y = x 2 d dx ( x 2 y ) = x 2 x 2 y = 1 x + C y = x + Cx 2
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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-09 JWDD027-Salas-v1 November 25, 2006 19:21 482 SECTION 9.1 9. y + 5 2 y = 1; H ( x ) = 5 2 dx = 5 2 x, integrating factor: e 5 x/ 2 e 5 x/ 2 y + 5 2 e 5 x/ 2 y = e 5 x/ 2 d dx e 5 x/ 2 y = e 5 x/ 2 e 5 x/ 2 y = 2 5 e 5 x/ 2 + C y = 2 5 + Ce 5 x/ 2 10. y y = 2 e x ; H ( x ) = dx, integrating factor: e x e x y e x y = 2 e 2 x d dx ( e x y ) = 2 e 2 x e x y = e 2 x + C y = e x + Ce x 11. y 2 y = 1 2 x ; H ( x ) = ( 2) dx = 2 x, integrating factor: e 2 x e 2 x y 2 e 2 x y = e 2 x 2 xe 2 x d dx e 2 x y = e 2 x 2 xe 2 x e 2 x y = 1 2 e 2 x + x e 2 x + 1 2 e 2 x + C = x e 2 x + C y = x + Ce 2 x 12. y + 2 x y = cos x x 2 ; H ( x ) = 2 x dx = 2ln | x | , integrating factor: x 2 x 2 y + 2 xy = cos x d dx [ x 2 y ] = cos x x 2 y = sin x + C y = sin x x 2 + C x 2 13. y 4 x y = 2 n ; H ( x ) = 4 x dx = 4 ln x = ln x 4 , integrating factor: e ln x 4 = x 4 x 4 y 4 x x 4 y = 2 nx 4 d dx x 4 y = 2 nx 4 x 4 y = 2 3 nx 3 + C y = 2 3 nx + Cx 4
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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-09 JWDD027-Salas-v1 November 25, 2006 19:21 SECTION 9.1 483 14. y + y = 2 + 2 x ; H ( x ) = dx, integrating factor: e x e x y + e x y = (2 + 2 x ) e x d dx ( e x y ) = 2(1 + x ) e x e x y = 2 xe x + C y = 2 x + Ce x 15. y e x y = 0; H ( x ) = e x dx = e x , integrating factor: e e x e e x y e x e e x y = 0 d dx e e x y = 0 e e x y = C y = Ce e x 16. y y = e x ; H ( x ) = dx, integrating factor: e x e x y e x y = 1 d dx ( e x y ) = 1 e x y = x + C y = xe x + Ce x 17. y + 1 1 + e x y = 1 1 + e x ; H ( x ) = 1 1 + e x dx = ln e x 1 + e x , integrating factor: e H ( x ) = e x 1 + e x e x 1 + e x y + 1 1 + e x · e x 1 + e x y = 1
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