ISM_T11_C09_C - 610 Chapter 9 Further Applications of...

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610 Chapter 9 Further Applications of Integration 14. y cx c 0 x y 2xy œÊ œ Ê œ Ê œ 22 y x y 2xy xx 24 2 w ± w y . So for the orthogonals: Êœ œ ± w 2y dy xd x 2 y x 2ydy xdx y C y C, ± Ê œ ± ² Ê œ ² 2 2 2 É C0 ³ 15. kx y 1 1 y kx k 2 2 1y x ²œÊ±œ Ê œ ± 2 2 0 2yx y 1 y 2x Ê ± œ ± x2 yy 1 y2 x x ab ˆ‰ w % ±± w a b y . So for the orthogonals: œ w 1y 2 x 2xy xy 2 dy x dx ln y C dy xy y dx 1 y y 2 2 x œ Ê œ Ê ±œ² ± ± 2 2 2 2 16. 2x y c 4x 2yy 0 y . For 222 4x 2x 2y y ²œÊ² œÊœ ±œ ± ww orthogonals: ln y ln x C dy y dy dx 2x y 2x dx œÊœÊ œ ² " # ln y ln x ln C y C x ² Ê œ 1/2 11 1/2 kk 17. y ce c œ Ê œ ! ± x y e x ey y e 1 e w x 2 e y ye y y. So for the orthogonals: ± Ê œ ± ±w ± w ydy dx x C dy y dx y 2 1 œÊ œ Ê œ² 2 y 2 xC y 2 xC Êœ² Êœ „ ² 2 È 18. y e ln y kx k 0 œÊœ Ê œ kx ln y x xy l n y x Š‹ 1 y 2 w ± y ln y 0 y . So for the orthogonals: ʱ œ Ê œ x yx yln y y ln y dy x dx dy dx y ln y x œ ± ± yln y y x C œ ± ² "" ## 1 4 y ln y x C œ ± ² y 2 1 2
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Section 9.5 Applications of First-Order Differential Equations 611 19. 2x 3y 5 and y x intersect at 1, 1 . Also, 2x 3y 5 4x 6y y 0 y y 1, 1 22 2 3 4x 2 6y 3 ±œ œ Ê ± œ Ê œ ² Ê œ ² ab ww w y x 2y y 3x y y 1, 1 . Since y y 1, the curves are orthogonal. 1 23 2 1 111 1 3x 3 2 3 2y 2 3 2 œÊ œ Êœ Ê œ †œ² œ ² www w w 2 1 ˆ‰ ˆ 20. (a) x dx y dy 0 C is the general equation Ê ± œ x y 2 2 of the family with slope y . For the orthogonals: w œ² x y y ln y ln x C or y C x w œÊ œ Ê œ ± œ yd y xyx dx 1 (where C e is the general equation of the 1 C œÑ orthogonals. (b) x dy 2y dx 0 2y dx x dy ²œ Êœ Ê œ dy 2y x dx ln y ln x C y C x is Ê œ ± Ê œ "" ## Š‹ dy yx dx 1 2 the equation for the solution family. ln y ln x C 0 y w ʲ œ Ê œ y2 y x 1 w slope of orthogonals is ² dy dx 2y x 2y dy x dx y C is the general ² Ê œ ² ± 2 x 2 2 equation of the orthogonals. 2 . y 4a 4ax and y 4b 4bx (at intersection) 4a 4ax 4b 4bx a b x a b "œ² œ±Ê ²œ± Ê ² œ± 2 2 2 2 a b a b a b x x a b. Now, y 4a 4a a b 4a 4a 4ab 4ab y 2 ab. ʱ ²œ± Êœ² œ ² ²œ ² ± œ Êœ È Thus the intersections are at a b, 2 ab . So, y 4a 4ax y which are equal to and È ²„ œ ² Êœ ² ² 1 4a 4a 2y 22 a b w È and at the intersections. Also, y 4b 4bx y which are equal to and ² œ ± Ê œ 4a a a 4b 4b 22a b b bb 2 y 2 È È ± w ÈÈ and at the intersections. y y . Thus the curves are orthogonal. 4b b b b aa 12 È ± œ²" ÉÉ abab
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612 Chapter 9 Further Applications of Integration CHAPTER 9 PRACTICE EXERCISES . y cos y dx 2tan y x C y tan Ê œ Ê œ ± Ê œ dy dy dx y cos y 2 2 1xC 2 ÈÈ È ˆ‰ 2 ± ² 2. y dy 3 x 1 dx y ln y x 1 C w ²± ± œÊœ ± Ê ² œ ± ± 3 yx1 y1 2 3 y ab 2 3. yy sec y sec x sec x dx tan x C sin y 2tan x C w œÊ œ Ê œ ± Ê œ ± 22 2 2 y dy sec y 2 sin y 1 2 2 4. y cos x dy sin x dx 0 y dy dx C y C 2 sin x 1 2 cos x 2 cos x cos x y 1 É ±œ Ê œ ²Ê œ ² ± Ê œ „± 2 2 ± 5. y xe x 2 e dy x x 2 dx e C e C ± ± ±² ± œ² Ê œ
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This note was uploaded on 10/13/2011 for the course MATHEMATIC 103 taught by Professor Thommas during the Spring '11 term at LCC Intl University.

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ISM_T11_C09_C - 610 Chapter 9 Further Applications of...

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