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  • CHAPTER 9 FIRST-ORDER DIFFERENTIAL EQUATIONS

    9.1 SOLUTIONS, SLOPE FIELDS AND EULER'S METHOD

    1. y x y slope of 0 for the line y x.w For x, y 0, y x y slope 0 in Quadrant I. w

    For x, y 0, y x y slope 0 in Quadrant III. w

    For y x , y 0, x 0, y x y slope 0 ink k k k w Quadrant II above y x. For y x , y 0, x 0, y x y slope 0 ink k k k w Quadrant II below y x. For y x , x 0, y 0, y x y slope 0 ink k k k w Quadrant IV above y x. For y x , x 0, y 0, y x y slope 0 ink k k k w Quadrant IV below y x. All of the conditions are seen in slope field (d).

    2. y y 1 slope is constant for a given value of y, slopew is 0 for y 1, slope is positive for y 1 and negative for y 1. These characteristics are evident in slope field (c).

    3. y slope 1 on y x and 1 on y x.w xy y slope 0 on the y-axis, excluding 0, 0 ,w xy a b and is undefined on the x-axis. Slopes are positive for x 0, y 0 and x 0, y 0 (Quadrants II and IV), otherwise negative. Field (a) is consistent with these conditions.

    4. y y x slope is 0 for y x and for y x.w 2 2

    For y x slope is positive and for y x slope isk k k k k k k k negative. Field (b) has these characteristics.

    Copyright 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

  • 538 Chapter 9 First-Order Differential Equations

    5. 6.

    7. y 1 t y t dt x y x ; y 1 1 t y t dt 1; x y, y 1 1 ' '1 1

    x 1dy dydx dxa b a b a b a b a ba b a b

    8. y dt ; y 1 dt 0; , y 1 0 ' '1 1

    x 11 1 1 1t dx x t dx x

    dy dya b a b

    9. y 2 1 y t sin t dt 1 y x sin x; y 0 2 1 y t sin t dt 2; 1 y sin x, ' '0 0

    x 0dy dydx dxa b a b a b a b a ba b a b a b

    y 0 2a b

    10. y 1 y t dt y x ; y 0 1 y t dt 1; y, y 0 1 ' '0 0

    x 0dy dydx dxa b a b a b a b a b

    11. y y 1 dx 1 1 (.5) 0.25," ! # yx 1!! y y 1 dx 0.25 1 (.5) 0.3,# " yx 2.50.25"" y y 1 dx 0.3 1 (.5) 0.75;$ # yx 30.3## y 1 P(x) , Q(x) 1 P(x) dx dx ln x ln x, x 0 v(x) e xdydx x x x k k" " "' ' ln x

    y x 1 dx C ; x 2, y 1 1 1 C 4 y " " # #x x 2 xx C x 4'

    #

    y(3.5) 0.6071 3.5 4 4.253.5 7#

    12. y y x (1 y ) dx 0 1(1 0)(.2) .2," ! ! ! y y x (1 y ) dx .2 1.2(1 .2)(.2) .392,# " " " y y x (1 y ) dx .392 1.4(1 .392)(.2) .5622;$ # # #

    x dx ln 1 y C; x 1, y 0 ln 1 C C ln 1 ydy1 yx x

    # # # # #" " " k k k k# #

    y 1 e y(1.6) .5416 a b1 x 2 #

    13. y y (2x y 2y ) dx 3 [2(0)(3) 2(3)](.2) 4.2," ! ! ! ! y y (2x y 2y ) dx 4.2 [2(.2)(4.2) 2(4.2)](.2) 6.216,# " " " " y y (2x y 2y ) dx 6.216 [2(.4)(6.216) 2(6.216)](.2) 9.6969;$ # # # #

    2y(x 1) 2(x 1) dx ln y (x 1) C; x 0, y 3 ln 3 1 C C ln 3 1dy dydx y k k # ln y (x 1) ln 3 1 y e e e 3e y(.6) 14.2765 # x 1 ln 3 1 ln 3 x 2x x x 2

    # #

    14. y y y (1 2x ) dx 1 1 [1 2( 1)](.5) .5," ! !# #!

    y y y (1 2x ) dx .5 (.5) [1 2( .5)](.5) .5,# " "# #"

    y y y (1 2x ) dx .5 (.5) [1 2(0)](.5) .625;$ # ## ##

    (1 2x) dx x x C; x 1, y 1 1 1 ( 1) C C 1 1 x xdyy y y# " "# # #

    y y(.5) 4 " " 1 x x 1 .5 (.5)# #

    Copyright 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

  • Section 9.1 Solutins, Slope Fields and Euler's Method 539

    15. y y 2x e dx 2 2(0)(.1) 2," ! ! x#

    !

    y y 2x e dx 2 2(.1) e (.1) 2.0202,# " " x 1# #

    "

    y y 2x e dx 2.0202 2(.2) e (.1) 2.0618,$ # # x 2#

    ##

    dy 2xe dx y e C; y(0) 2 2 1 C C 1 y e 1 y(.3) e 1 2.0942 x x x 3# # # #

    16. y y y e dx 2 2 e (.5) 3," ! ! a b a bx! 0 y y y e dx 3 3 e (.5) 5.47308,2 1 1 0.5 a b a bx1 y y y e dx 5.47308 5.47308 e (.5) 12.9118,3 2 2 1.0 a b a bx2 y e e dx ln y e C; x 0, y 2 ln 2 1 C C ln 2 1 ln y e ln 2 1dy dydx y

    x x x x l l l l

    y 2e y 1.5 2e 65.0292 e ex 1.5" "a b

    17. y 1 1(.2) 1.2," y 1.2 (1.2)(.2) 1.44,# y 1.44 (1.44)(.2) 1.728,$ y 1.728 (1.728)(.2) 2.0736,% y 2.0736 (2.0736)(.2) 2.48832;&

    dx ln y x C y Ce ; y(0) 1 1 Ce C 1 y e y(1) e 2.7183dyyx x "

    !

    18. y 2 (.2) 2.4," 21 y 2.4 (.2) 2.8,# 2.41.2 y 2.8 (.2) 3.2,$ 2.81.4 y 3.2 (.2) 3.6,% 3.21.6 y 3.6 (.2) 4;& 3.61.8 ln y ln x C y kx; y(1) 2 2 k y 2x y(2) 4dyy x

    dx

    19. y 1 (.5) .5," ( 1)

    1

    #

    y .5 (.5) .39794,# ( .5)

    1.5

    #

    y .39794 (.5) .34195,$ ( .39794)

    2

    #

    y .34195 (.5) .30497,% ( .34195)

    2.5

    #

    y .27812, y .25745, y .24088, y .2272;& ' ( )

    2 x C; y(1) 1 1 2 C C 1 y y(5) .2880dyy ydx

    x 1 x 1 5#

    " " "# #

    20. y 1 0 sin 1 1," " a b 3 y 1 sin 1 1.09350,# " " 3 3 y 1.09350 sin 1.09350 1.29089,$ " 23 3 y 1.29089 sin 1.29089 1.61125,% " 33 3 y 1.61125 sin 1.61125 2.05533,& " 43 3 y 2.05533 sin 2.05533 2.54694;' " 53 3 y x sin y csc y dy x dx ln csc y cot y x C csc y cot y e Cew l l 12

    2 x C x1 12 22 2

    Ce cot Ce ; y 0 1 cot Ce C cot cot e 1 cos y y ysin y 2 2 2 2x x 0 x1 1 1 1 12 2 2

    2 2 2 a b y 2 cot cot e , y 2 2 cot cot e 2.65591 1 x 1 21 12 2 a b

    12

    2

    Copyright 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

  • 540 Chapter 9 First-Order Differential Equations

    21. y 1 x 1 x y e y x 1 x 1 x y e 1 x 1 x y 1 y a b a b a b a ba b0 0 0 0 0 0 0 0 0 0x x x x 0 0 0 1 1 x y e y 1 x 1 x y e x x ydy dy dydx dx dx0 0 0 0

    x x x x a b a b 0 0

    22. y f x , y x y y f t dt C, y x f t dt C C C y y f t dt yw a b a b a b a b a b a b0 0 0 0 0x x xx x x' ' '0 0 0

    0

    23-34. Example CAS commands: :Maple ode := diff( y(x), x ) = y(x); icA := [0, 1]; icB := [0, 2]; icC := [0,-1]; DEplot( ode, y(x), x=0..2, [icA,icB,icC], arrows=slim, linecolor=blue, title="#23 (Section 9.1)" ); :Mathematica To plot vector fields, you must begin by loading a graphics package.

  • Section 9.1 Solutins, Slope Fields and Euler's Method 541

    23. 24.

    25. 26.

    27. 28.

    35. 2xe , y 0 2 y y 2x e dx y 2x e 0.1 y 0.2x edydx n 1 n n n n n n x x x x2 2 2n n na b a b 2

    On a TI-84 calculator home screen, type the following commands: 2 STO > y: 0 STO > x: y (enter) y 0.2*x*e^(x^2) STO > y: x 0.1 STO > x: y (enter, 10 times) The last value displayed gives y 1 3.45835Eulera b The exact solution: dy 2xe dx y e C; y 0 2 e C C 1 y 1 e x x x

    2 2 2a b 0 y 1 1 e 3.71828 exacta b

    36. 2y x 1 , y 2 y y 2y x 1 dx y 0.2 y x 1dydx 22 2 21

    n 1 n n n nn n a b a b a b a b On a TI-84 calculator home screen, type the following commands: 0.5 STO > y: 2 STO > x: y (enter)

    y 0.2*y x 1 STO > y: x 0.1 STO > x: y (enter, 10 times) 2a b The last value displayed gives y 2 0.19285Eulera b The exact solution: 2y x 1 2x 2 dx x 2x C x 2x Cdy dydx y y y

    2 2 21 1 a b a b2 y 2 2 2 2 C C C 2 x 2x 2 ya b a b a b 1 1 1 12 1 2 y x 2x 22 2 2 y 3 0.2a b 1

    3 2 3 2 a b a b2

    Copyright 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

  • 542 Chapter 9 First-Order Differential Equations

    37. , y 0 y 0 y y dx y 0.1 y 0.dydx y y y yx

    n 1 n n nx x x

    " " a b a b n n n

    n n n

    On a TI-84 calculator home screen, type the following commands: 1 STO > y: 0 STO > x: y (enter)

    y 0.1*( x /y) STO > y: x 0.1 STO > x: y (enter, 10 times) The last value displayed gives y 1 1.5000Eulera b The exact solution: dy dx y dy x dx x C; 0 C C

    a ba bxy 2 3 2 2 2 3 2

    y 2 1 1 2 13/2 y 0 3/2 a b2 2 2

    x y x 1 y 1 1 1.5275 " y2 3 2 3 32 1 4 43/2 3/2

    exact3/22 a b a b

    38. 1 y , y 0 0 y y 1 y dx y 1 y 0.1 y 0.1 1 ydydx2 2 2 2

    n 1 n n nn n n a b a b a ba b a b On a TI-84 calculator home screen, type the following commands: 0 STO > y: 0 STO > x: y (enter)

    y 0.1*(1 y ) STO > y: x 0.1 STO > x: y (enter, 10 times) 2

    The last value displayed gives y 1 1.3964Eulera b The exact solution: dy 1 y dx dx tan y x C; tan y 0 tan 0 0 0 C C 0 a b a b2 1 1 1dy1 y 2 tan y x y tan x y tan 1 1.5574 " 1 exacta b

    39. Example CAS commands: :Maple ode := diff( y(x), x ) = x + y(x);ic := y(0)=-7/10; x0 := -4;x1 := 4;y0 := -4; y1 := 4; b := 1; P1 := DEplot( ode, y(x), x=x0..x1, y=y0..y1, arrows=thin, title="#39(a) (Section 9.1)" ): P1; Ygen := unapply( rhs(dsolve( ode, y(x) )), x,_C1 ); # (b) P2 := seq( plot( Ygen(x,c), x=x0..x1, y=y0..y1, color=blue ), c=-2..2 ): # (c) display( [P1,P2], title="#39(c) (Section 9.1)" ); CC := solve( Ygen(0,C)=rhs(ic), C ); # (d) Ypart := Ygen(x,CC); P3 := plot( Ypart, x=0..b, title="#39(d) (Section 9.1)" ): P3; euler4 := dsolve( {ode,ic}, numeric, method=classical[foreuler], stepsize=(x1-x0)/4 ): # (e) P4 := odeplot( euler4, [x,y(x)], x=0..b, numpoints=4, color=blue ): display( [P3,P4], title="#39(e) (Section 9.1)" ); euler8 := dsolve( {ode,ic}, numeric, method=classical[foreuler], stepsize=(x1-x0)/8 ): # (f) P5 := odeplot( euler8, [x,y(x)], x=0..b, numpoints=8, color=green ): euler16 := dsolve( {ode,ic}, numeric, method=classical[foreuler], stepsize=(x1-x0)/16 ): P6 := odeplot( euler16, [x,y(x)], x=0..b, numpoints=16, color=pink )

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