# Calculus: One and Several Variables

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-19 JWDD027-Salas-v1 January 4, 2007 19:13 984 SECTION 19.1 CHAPTER 19 SECTION 19.1 1. y 0 + xy = xy 3 = y 3 y 0 + xy 2 = x. Let v = y 2 ,v 0 = 2 y 3 y 0 . 1 2 v 0 + xv = x v 0 2 xv = 2 x e x 2 v 0 2 xe x 2 v = 2 xe x 2 e x 2 v = e x 2 + C v =1+ Ce x 2 y 2 = 1 1+ x 2 . 2. y 0 y = ( x 2 + x +1) y 2 = y 2 y 0 y 1 = ( x 2 + x . Let v = y 1 0 = y 2 y 0 . v 0 v = ( x 2 + x v 0 + v = x 2 + x +1 e x v = Z e x ( x 2 + x dx = x 2 e x xe x +2 e x + C v = x 2 x +2+ x y = 1 x 2 x x . 3. y 0 4 y =2 e x y 1 2 = y 1 2 y 0 4 y 1 2 e x . Let v = y 1 2 0 = 1 2 y 1 2 y 0 . 2 v 0 4 v e x v 0 2 v = e x e 2 x v 0 2 e 2 x v = e x e 2 x v = e x + C v = e x + 2 x y =( 2 x e x ) 2 . 4. y 0 = 1 2 xy + y 2 x = yy 0 1 2 x y 2 = 1 2 x . Let v = y 2 0 0 . 1 2 v 0 1 2 x v = 1 2 x v 0 1 x v = 1 x 1 x v 0 1 x 2 v = 1 x 2 1 x v = 1 x + C v = Cx 1 y 2 = 1 .

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-19 JWDD027-Salas-v1 January 4, 2007 19:13 SECTION 19.1 985 5. ( x 2) y 0 + y =5( x 2) 2 y 1 2 = y 1 2 y 0 + 1 x 2 y 1 2 x 2) . Let v = y 1 2 ,v 0 = 1 2 y 1 2 y 0 . 2 v 0 + 1 x 2 v x 2) v 0 + 1 2( x 2) v = 5 2 ( x 2) x 2 v 0 + 1 2 x 2 v = 5 2 ( x 2) 3 2 x 2 v =( x 2) 5 2 + C v x 2) 2 + C x 2 y = · ( x 2) 2 + C x 2 ¸ 2 . 6. yy 0 xy 2 + x =0 . Let v = y 2 0 =2 0 . 1 2 v 0 xv = x v 0 2 xv = 2 x e x 2 v 0 2 xe x 2 v = 2 xe x 2 e x 2 v = e x 2 + C v =1+ Ce x 2 y = 1+ x 2 . 7. y 0 + xy = y 3 e x 2 = y 3 y 0 + xy 2 = e x 2 . Let v = y 2 0 = 2 y 3 y 0 . 1 2 v 0 + xv = e x 2 v 0 2 xv = 2 e x 2 e x 2 v 0 2 xe x 2 v = 2 e x 2 v = 2 x + C v = 2 xe x 2 + x 2 y 2 = x 2 2 xe x 2 . C =4 = y 2 =4 e x 2 2 xe x 2 . 8. y 0 + 1 x y = ln x x y 2 = y 2 y 0 + 1 x y 1 = ln x x . Let v = y 1 0 = y 2 y 0 . v 0 + 1 x v = ln x x v 0 1 x v = ln x x 1 x v 0 1 x 2 v = ln x x 2
P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-19 JWDD027-Salas-v1 January 4, 2007 19:13 986 SECTION 19.1 1 x v = Z ln x x 2 dx = 1 x (ln x +1)+ C v =ln x +1+ Cx y = 1 ln x . 1= 1 ln1+1+ C = C =0 = y = 1 ln x +1 . 9. 2 x 3 y 0 3 x 2 y = y 3 = y 3 y 0 3 2 x y 2 = 1 2 x 3 . Let v = y 2 ,v 0 = 2 y 3 y 0 . 1 2 v 0 3 2 x v = 1 2 x 3 v 0 + 3 x v = 1 x 3 x 3 v 0 +3 x 2 v = 1 x 3 v = x + C v = C x x 3 y 2 = x 3 C x 1 C x = C =2 = y 2 = x 3 2 x . 10. y 0 + tan xy = y 2 sec 3 x = y 2 y 0 + tan xy 1 = sec 3 x. Let v = y 1 0 = y 2 y 0 . v 0 + tan xv = sec 3 x v 0 tan xv = sec 3 x cos xv 0 sin = sec 2 x cos = tan x + C cos x y = tan x + C cos0 3 = tan0 + C = C = 1 3 = cos x y = 1 3 tan x. 11. y 0 y x ln y = xy = y 0 y 1 x ln y = x. Let u y, u 0 = y 0 y . u 0 1 x u = x 1 x u 0 1 x 2 u =1 1 x u = x + C u = x 2 + ln y = x 2 + Cx.

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-19 JWDD027-Salas-v1 January 4, 2007 19:13 SECTION 19.1 987 12. (a) y 0 + yf ( x )ln y = g ( x ) y y 0 y + f ( x y = g ( x ) u 0 + f ( x ) u = g ( x ) .
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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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ch19[1] - P1 PBU/OVY JWDD027-19 P2 PBU/OVY JWDD027-Salas-v1...

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