Calculus: One and Several Variables

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Unformatted text preview: 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 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 + xe − 2 x + 1 2 e − 2 x + C = xe − 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 P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-09 JWDD027-Salas-v1...
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This note was uploaded on 04/30/2011 for the course MATH 1431 taught by Professor Any during the Spring '08 term at University of Houston.

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Ch09[1] - P1 PBU/OVY P2 PBU/OVY QC PBU/OVY T1 PBU JWDD027-09 JWDD027-Salas-v1 19:21 SECTION 9.1 481 CHAPTER 9 SECTION 9.1 1 y 1 x = 1 2 e x 2 2 y 1

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