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Homework-8_exact_Equations_0001

# Homework-8_exact_Equations_0001 - EX E RC i S ES 2 4...

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Unformatted text preview: EX E RC i S ES 2 . 4 Answers to selected odd-numbered problems begin on page ANS—2. In Problems 1—20 determine whether the given differential 2 3' 3 . _ = equation is exact. Ifit is exact, solve it. 12- (3x 3/ + 6 )dx + (x + xe" 2y) dy 0 1. (2x—1)dx+(3y+7)dy=0 13.xd_)’_m _y+6x2 2. (2x+y)dx—(x+6y)dy=0 dx 3.(5x+4y)dx+(4x—8y3)dy=0 14.(1_§+x)ﬂ+y:§_1 4.(siny—ysindex+(cosx+xcosy¥y)dy=0 y' dx x i 5. (2xy2—3)dx+(2x2y+4)dy=0 15__(x2y3_+ 1 )dx+ +x3y _0 1+ 9x2 dy 1 6. (2y — — + cos 3x) 3—)) + —5 — 4x3 + 3y sin 3): = 0 x x x 16. (Sy-n2x)y’—2y=0 2_ 2 2— = 7- (x 3’ )dx+ (x 2xy)dy 0 l7. (tanx— sinxsiny)dx+cosxcosydy=0 8. (1 + lnx + E) dx = {1 — In x) dy 18. (23) sin x cosx —— y + 2y2exy1)dx = _ . 2 _ x 1 9. {x — 3:3 + ygsinxhix = (3x);2 + 2ycosx)dy .(x 5111 x 4xye Udy 10. {x3 + y3) dx + 3x3;2 dy = 0 19. (4:33; - 15f2 — y) dz + (t4 + 3y2 — t) dy = 0 _ 1 1 1 y 11.(ylnyv-e*")dx+(—+xln )d =0 2_(— +—— ) +( :'+ ) = y 3’ J’ 0 r :2 {2+3}? dt ye {2+3}: dy O In Problems 21—25 501% the given initial-value problem. In Problems 31—36 solve the given differential equation by 21. (x + 302 dx + (2x); + x2 _ 1) dy = 0’ 3’0) = 1 ﬁnding, as in Example 4, an appropriate integrating factor. 2 _ 22. (e‘+y)dx+ (2 +x+yendy= 0, y(0) =1 31- (23’ + 3’0“?” 2"ny “ 0 '23. (4y-‘r 2: — 5)d:+ (6y +4r—1)dy= 0,. y{-1) = 2 32- 3’06 +3) +1)dx + (x + 2y)dy= 0 33.6 d +4 +92d =0 3y2_32 dy I xyx (y x))’ 24. 5 —+—4:0a 3’0): 3) dt 2y 2 34. cosxa‘x + 1+ — sinxdy = 0 3’ 25. (3:2 cosx—3x2y— 2x)dx +(2ysinx—x3+lny)dy= 0, y(0)=e 1-. 2 3 2_ 3 ' _ 26. (I :y2+C03x—2xy):—:=y(y+sinx), 31(0) =1 36- (J) +xy)dx+(5y xy+y smy)dy—0 35. (10 — 6y+ e_3x] dx - 2dy= 0 In Problems 3'? and 38 solve the given initial-value problem In Problems 27 and 28 ﬁnd the value of k so that the given by ﬁnding, as in Example 4, an appropriate integrating factor. differential equation is exact. - 37. x dx + (xzy + 4y) d)’ = 0. M4} : 27. (3:3 + kxy“ — 2x) dx + (3x312 + 20x2y3) dy = 0 38. (x2 + 3:2 — 5) dx = (y + xy) dy, y(0} = 1 28. (6x);3 + cos 3)) dx + (2M2);2 — 3: sin y) dy = 0 39. (a) Show that a one-parameter family of solutions of In Problems 29- and 30 verify that the given differential equa- the equation tion is not exact. Multiply the given differential equation by the indicated integrating factor ,u(x, y) and verify that the (4xy + 336*) dx + (2y + 2 x2) 51), = 0 new equation is exact. Solve. _ is x3 + 2x2); + y2 = 29. ("xy sm x + 2y cos x) dx + 2x cos x dy = 0; MI. 3)) = xy 30. (x2 + ny - yz) dx + (y: + 2x31 — x2) dy = 0; 9 M17. 30 = (x + y)"2 ...
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