1. 1 1 , , , , , , / 17 2013,

2. dy = f (x, y) dxy = f (x, y) 3. dy = f (x, y) dxy = f (x, y) 4. dy = f (x, y) dx f (x, y) = f (x)y = f (x).y = f (x, y) 5. dy = f (x, y) dx f (x, y) = f (x)y = f (x, y)y = f (x). y (x) d =f (x) dx + C, y(x) =f (x) dx + C. 6. 2y(0) = 1.y = ex , xy(x) =02es ds + 1. 7. 9yy = 4x. 9y(x)y (x)dx + 4xdx = C 8. 9yy = 4x. 9 9y(x)y (x)dx + 4xdx = Cydy + 4x2 = C 9y2 + 4x2 = 2C 9. 9yy = 4x. 9 9y(x)y (x)dx + 4xdx = Cydy + 4x2 = C 9y2 + 4x2 = 2C :y2 x2 4 + 9 =C 10. y + yx = x. 11. y + yx = x. y1y=x 12. y + yx = x. y1y=xdy = 1yxdx 13. y + yx = x. y1y=x ln |1 y| =x2 +C 2dy = 1yxdx 14. y + yx = x. y1y=x ln |1 y| =x2 +C 2dy = 1y xdx x2y = 1 + Ce 2 15. y + yx = x. y1y=x ln |1 y| =x2 +C 2dy = 1y xdx x2y = 1 + Ce 2 16. dy = f (x, y) dxy = f (x, y) 17. dy = f (x, y) dx f (x, y) = f (y)y = f (x, y)dy dx 1 = f (y) = . dx dy f (y) x(y) = y = ky y = Cekx . dy 1 dx = y2 y = Cx1 dy + C. f (y)y = 0. 18. et/2 2 10 ; x = et/2 x(t) = 2et/2 + C. 0 = x(0) = 2e0/2 + C = 2 + C C = 2. x(t) = et/2 2. x(2) = 2e2/2 2 3.44 , x(10) = 2e10/2 2 294 . 19. t = 10 t2 sm 2 t = 0 1 10 m ; s x = t2 , x = vx(0) = 1, v = t2 ,x (0) = 10.v(0) = 10, v, x. 20. y = f (x, y) y (x, y). 21. y = f (x, y) y (x, y). : y = xy y(0) = 0.2, y(0) = 0, y(0) = 0.2. 22. y = f (x, y) y (x, y). : y = xy y(0) = 0.2, y(0) = 0, y(0) = 0.2. -3-2-1012333221100-1-1-2-2-3-3 -3-2-10123 23. y = f (x, y) y (x, y). : y = xy y(0) = 0.2, y(0) = 0, y(0) = 0.2. -3-2-10123-3-2-1012333 3322 2211 1100 00-1-1-1-1-2-2-2-2-3-3-3 -3-2-10123-3 -3-2-10123 24. / ! 25. y = f (x, y),y(x0 ) = y0 .(i) ; (ii) ( ); 26. y = f (x, y),y(x0 ) = y0 .(i) ; (ii) ( ); : y = 1, y(0) = 0 y = ln |x| + C. x 27. Picard f (x, y) f y (x0 , y0 ), y = f (x, y),y(x0 ) = y0 , ( x) . 28. Picard f (x, y) f y (x0 , y0 ), y = f (x, y),y(x0 ) = y0 , ( x) . 1. y = 1 , y(0) = 0 x 2. y = 2 |y|, y(0) = 0 29. : y = 1 , xy(0) = 0 y = ln |x| + C. 30. : y = 1 , xy(0) = 0 y = ln |x| + C. : y = -3-2-1012333221100-1-1-2-2-3-3 -3-2-101231 x y = 2 |y|. 31. : y = 1 , xy(0) = 0 y = ln |x| + C. : y = -3-2-1012-33-21 x y = 2 |y|. -1012333 3322 2211 1100 00-1-1-1-1-2-2-2-2-3-3-3 -3-2-10123-3 -3-2-10123 32. y = y2 ,y(0) = A, 33. y = y2 , x =y(0) = A, 1 , y2 34. y = y2 ,y(0) = A,x = x=1 , y21y+ C, 35. y = y2 ,y(0) = A,x = x=1 , y21 y=y+ C,1 Cx. 36. y = y2 ,y(0) = A,x = x=1y y= 1 , y2 + C,1 Cxy(0) = A C =.1 y= A11 A x. 37. y = f (x, y), 38. y = f (x, y), f (x, y) = f (x)g(y), 39. y = f (x, y), f (x, y) = f (x)g(y), y = f (x)g(y), 40. y = f (x, y), f (x, y) = f (x)g(y), y = f (x)g(y), dy dy = f (x)g(y) = f (x) dx dx g(y)dy = g(y)f (x) dx + C. 41. y = xy. 42. y = xy. y = 0. 43. y = xy. y = 0. dy = yx dx + C. 44. y = xy. y = 0. dy = yx dx + C.ln |y| =x2 +C 2 45. y = xy. y = 0. dy = yx dx + C.ln |y| =x2 +C 2222x x x |y| = e 2 +C = e 2 eC = De 2, 46. y = xy. y = 0. dy = yx dx + C.ln |y| =x2 +C 2222x x x |y| = e 2 +C = e 2 eC = De 2 x2y = De 2D., 47. dy = f (x)g(y) dx 48. dy = f (x)g(y) dx y = y(x) x. x!dy dx 49. dy = f (x)g(y) dx y = y(x) x. x! 1 dy = f (x) g(y) dxdy dx 50. dy = f (x)g(y) dx y = y(x) x. x! 1 dy = f (x) g(y) dx x. 1 dy dx = g(y) dxf (x) dx + C.dy dx 51. dy = f (x)g(y) dx y = y(x) x. x! 1 dy = f (x) g(y) dx x. 1 dy dx = g(y) dxf (x) dx + C. 1 dy = g(y)f (x) dx + C.dy dx 52. y =xyy2 + 1. 53. y =xyy2 + 11 y2 + 1 dy = y + y y. dy = x dx 54. y =xyy2 + 11 y2 + 1 dy = y + y y. dy = x dxy2 x2 + ln |y| = +C 2 2 55. y =xyy2 + 11 y2 + 1 dy = y + y y. dy = x dxy2 x2 + ln |y| = +C 2 2 y2 + 2 ln |y| = x2 + C. . ! 56. y =xyy2 + 11 y2 + 1 dy = y + y y. dy = x dxy2 x2 + ln |y| = +C 2 2 y2 + 2 ln |y| = x2 + C. . ! y 2y +2 = 2x. y 57. y =xyy2 + 11 y2 + 1 dy = y + y y. dy = x dxy2 x2 + ln |y| = +C 2 2 y2 + 2 ln |y| = x2 + C. . ! y 2y +2 = 2x. y y(x) = 0. 58. x2 y = 1 x2 + y2 x2 y2 , y(1) = 0 59. x2 y = 1 x2 + y2 x2 y2 , y(1) = 0 x2 y = (1 x2 )(1 + y2 ). 60. x2 y = 1 x2 + y2 x2 y2 , y(1) = 0 x2 y = (1 x2 )(1 + y2 ). 1 x2 y 1 = 1, x2 1 + y2 x2 1 1 x + C, y = tan x+C . arctan(y) = x x y= 1 + y2 , 0 = tan(2 + C) C = 2 ( 2 + , . . . ) y = tan1xx+2. 61. (100o C). 1 T 95o C. A 10o C 70o C. ; 62. (100o C). 1 T 95o C. A 10o C 70o C. ; dT = k(A T), A = 10, T(0) = 100, T(1) = 95. dt 63. (100o C). 1 T 95o C. A 10o C 70o C. ; dT = k(A T), A = 10, T(0) = 100, T(1) = 95. dt 1dT =k A T dt 64. (100o C). 1 T 95o C. A 10o C 70o C. ; dT = k(A T), A = 10, T(0) = 100, T(1) = 95. dt 1dT = k ln(AT) = kt+C AT = D ekt , T = AD ekt A T dt T = 10 D ekt 100 = T(0) = 10 D D = 90 65. (100o C). 1 T 95o C. A 10o C 70o C. ; dT = k(A T), A = 10, T(0) = 100, T(1) = 95. dt 1dT = k ln(AT) = kt+C AT = D ekt , T = AD ekt A T dt T = 10 D ekt 100 = T(0) = 10 D D = 90 T = 10 + 90 ekt 95 = T(1) = 10 + 90 ek k = ln 959010 0.06. 66. (100o C). 1 T 95o C. A 10o C 70o C. ; dT = k(A T), A = 10, T(0) = 100, T(1) = 95. dt 1dT = k ln(AT) = kt+C AT = D ekt , T = AD ekt A T dt T = 10 D ekt 100 = T(0) = 10 D D = 90 T = 10 + 90 ekt 95 = T(1) = 10 + 90 ek k = ln 959010 0.06.70 = 10 + 90e0.06t t = ln 709010 0.06 6, 8 . 67. y + p(x)y = f (x).(1) 68. y + p(x)y = f (x).(1) p(x) f (x) p(x) f (x) 69. y + p(x)y = f (x).(2) 70. y + p(x)y = f (x). r (x)y + r (x)p(x)y =d r (x)y . dx(2) 71. y + p(x)y = f (x). r (x)y + r (x)p(x)y =d r (x)y . dxd r (x)y = r (x)f (x). dx (2) 72. y + p(x)y = f (x). r (x)y + r (x)p(x)y =d r (x)y . dxd r (x)y = r (x)f (x). dx y y(2) 73. y + p(x)y = f (x). r (x)y + r (x)p(x)y =d r (x)y . dxd r (x)y = r (x)f (x). dx y y r (x)!(2) 74. r (x) , p(x). 75. r (x) , p(x). r (x) = e p(x)dx 76. r (x) , p(x). r (x) = e p(x)dx . y + p(x)y = f (x), e p(x)dx y + e p(x)dx p(x)y = e p(x)dx f (x), d e p(x)dx y = e p(x)dx f (x), dx e p(x)dx y = e p(x)dx f (x) dx + C, y = ep(x)dxep(x)dx f (x) dx + C. 77. y + p(x)y = f (x). 78. y + p(x)y = f (x). r (x)y + r (x)p(x)y =d r (x)y . dx 79. y + p(x)y = f (x). r (x)y + r (x)p(x)y =d r (x)y . dxr (x) = p(x)r (x) r (x) = ep(x)dx 80. y + p(x)y = f (x). r (x)y + r (x)p(x)y =d r (x)y . dxr (x) = p(x)r (x) r (x) = eep(x)dxy + p(x)y = f (x), e p(x)dx f (x),p(x)dx y + e p(x)dx p(x)y =d e p(x)dx y = e p(x)dx f (x), dx e p(x)dx y = e p(x)dx f (x) dx + Cy = ep(x)dxep(x)dx f (x) dx + C. 81. y + 2xy = exx2y(0) = 1. 82. y + 2xy = exx xx2p(x) = 2x, f (x) = e2, r (x) = ey(0) = 1. p(x) dx = ex2 . 83. y + 2xy = exx xx2p(x) = 2x, f (x) = e22, r (x) = e 2y(0) = 1. p(x) dx = ex2 . 22ex y + 2xex y = exx ex , d x2 e y = ex . dx 84. y + 2xy = exx xx2p(x) = 2x, f (x) = e22y(0) = 1., r (x) = ep(x) dx = ex2 .222ex y + 2xex y = exx ex , d x2 e y = ex . dx 2ex y = ex + C, 22y = exx + Cex . 85. y + 2xy = exx xx2p(x) = 2x, f (x) = e22y(0) = 1., r (x) = ep(x) dx = ex2 .222ex y + 2xex y = exx ex , d x2 e y = ex . dx 2ex y = ex + C, 22y = exx + Cex . 1 = y(0) = 1 + C C = 2. 86. y + 2xy = exx xx2p(x) = 2x, f (x) = e22y(0) = 1., r (x) = ep(x) dx = ex2 .222ex y + 2xex y = exx ex , d x2 e y = ex . dx 2ex y = ex + C, 22y = exx + Cex . 1 = y(0) = 1 + C C = 2. 22y = exx 2ex . 87. f (x) dx + C xf (t) dt + C. x0 88. y + p(x)y = f (x),y(x0 ) = y0 . 89. y + p(x)y = f (x), y(x) = e xx0p(s) dsxy(x0 ) = y0 . te x0 x0p(s) dsf (t) dt + y0 . . 90. 100 10 60 . 0.1/ 5 3 . ; 91. 100 10 60 . 0.1/ 5 3 . ; x(t) : t x ( )t ( )t. 92. 100 10 60 . 0.1/ 5 3 . ; x(t) : t x ( )t ( )t. dx = ( ) dt ( ). 93. 100 10 60 . 0.1/ 5 3 . ; x(t) : t x ( )t ( )t. dx = ( ) dt ( ). dx x dx 3 = (5 0.1) 3 + x = 0.5. dt 60 + 2t dt 60 + 2t 94. () dx 3 + x = 0.5. dt 60 + 2t 95. ()r (t) = expdx 3 + x = 0.5. dt 60 + 2t 3 3 dt = exp ln(60 + 2t) = (60 + 2t)3/2 . 60 + 2t 2 96. ()r (t) = expdx 3 + x = 0.5. dt 60 + 2t 3 3 dt = exp ln(60 + 2t) = (60 + 2t)3/2 . 60 + 2t 2 (60 + 2t)3/2 dx + (60 + 2t)3/2dt3 x = 0.5(60 + 2t)3/2 , 60 + 2td 3/2 x = 0.5(60 + 2t)3/2 , (60 + 2t) dt (60 + 2t)3/2 x =0.5(60 + 2t)3/2 dt + C, 97. ()r (t) = expdx 3 + x = 0.5. dt 60 + 2t 3 3 dt = exp ln(60 + 2t) = (60 + 2t)3/2 . 60 + 2t 2 (60 + 2t)3/2 dx + (60 + 2t)3/2dt3 x = 0.5(60 + 2t)3/2 , 60 + 2td 3/2 x = 0.5(60 + 2t)3/2 , (60 + 2t) dt (60 + 2t)x = (60 + 2t)3/2 x = (60 + 2t)3/23/2 x =(60 + 2t)3/220.5(60 + 2t)3/2 dt + C,dt + C(60 + 2t)3/2 ,1 (60 + 2t)5/2 + C(60 + 2t)3/2 10 98. ()r (t) = expdx 3 + x = 0.5. dt 60 + 2t 3 3 dt = exp ln(60 + 2t) = (60 + 2t)3/2 . 60 + 2t 2 (60 + 2t)3/2 dx + (60 + 2t)3/2dt3 x = 0.5(60 + 2t)3/2 , 60 + 2td 3/2 x = 0.5(60 + 2t)3/2 , (60 + 2t) dt (60 + 2t)x = (60 + 2t)3/2 x = (60 + 2t)3/23/2 x =(60 + 2t)3/220.5(60 + 2t)3/2 dt + C,dt + C(60 + 2t)3/2 ,1 (60 + 2t)5/2 + C(60 + 2t)3/2 10 99. () 100 10 60 . 0.1/ 5 3 . ; x(t) =60 + 2t + C(60 + 2t)3/2 . 10 100. () 100 10 60 . 0.1/ 5 3 . ; x(t) =60 + 2t + C(60 + 2t)3/2 . 1010 = x(0) = 6 + C(60)3/2 C = 4(603/2 ) 1859.03 101. () 100 10 60 . 0.1/ 5 3 . ; x(t) =60 + 2t + C(60 + 2t)3/2 . 1010 = x(0) = 6 + C(60)3/2 C = 4(603/2 ) 1859.03 60 + 2t = 100, . t = 20 x(20) 10 + 1859.03(100)3/2 11.86. 102. y = (x y + 1)2 . 103. y = (x y + 1)2 . v=xy+1 v =1y, y =1v. 104. y = (x y + 1)2 . v=xy+1 v =1y, y =1v. 1v = v2 v = 1v2 11vdv = dx 21 v+1 = x+C, ln 2 v1 105. y = (x y + 1)2 . v=xy+1 v =1y, y =1v. 1v = v2 v = 1v2 11vdv = dx 21 v+1 = x+C, ln 2 v1v+1 v+1 = De2x = e2x+2C v1 v1 106. y = (x y + 1)2 . v=xy+1 v =1y, y =1v. 1v = v2 v = 1v2 11vdv = dx 21 v+1 = x+C, ln 2 v1v+1 v+1 = De2x = e2x+2C v1 v1 xy+2 xy =De2x y = x, y = x + 2. 107. y = (x y + 1)2 . v=xy+1 v =1y, y =1v. 1v = v2 v = 1v2 11vdv = dx 21 v+1 = x+C, ln 2 v1v+1 v+1 = De2x = e2x+2C v1 v1 xy+2 xy =De2x y = x, y = x + 2.x y + 2 = (x y)De2xy=Dxe2x x 2 . De2x 1 108. yy y2 y (cos y)y (sin y)y y ey y2 y3 sin y cos y ey 109. Bernoulliy + p(x)y = q(x)yn . : v = y1n 110. xy + y(x + 1) + xy5 = 0, v = y15 = y4 ,y(1) = 1.v = 4y5 yy54v =y. 111. xy + y(x + 1) + xy5 = 0, v = y15 = y4 ,y(1) = 1.v = 4y5 yxy + y(x + 1) + xy5 = 0, xy54v + y(x + 1) + xy5 = 0,x4v + y4 (x + 1) + x = 0, x v + v(x + 1) + x = 0, 4v4(x + 1) v = 4. xy54v =y. 112. () v4(x + 1) v = 4. x 113. () v r (x) = exp4(x + 1)x4(x + 1) v = 4. x dx = e4x4 ln(x) = e4x x4 =e4x . x4 114. () v r (x) = exp4(x + 1)x d dx4(x + 1) v = 4. x dx = e4x4 ln(x) = e4x x4 =e4x e4x v =4 4 , x4 x x e4s e4x v= 4 4 ds + 1, 4 x s 1 x e4s v = e4x x4 4 ds + 1 . s4 1e4x . x4 115. () v r (x) = exp4(x + 1)x d dx4(x + 1) v = 4. x dx = e4x4 ln(x) = e4x x4 =e4x e4x v =4 4 , x4 x x e4s e4x v= 4 4 ds + 1, 4 x s 1 x e4s v = e4x x4 4 ds + 1 . s4 1y4 = e4x x4 4 y=x e4ss41ds + 1 ,ex x 4x e4sds + 11/4.e4x . x4 116. y =Fy . x v=y xy = v + xv . 117. y =Fy . x v= v+xv = F(v)y x y = v + xv .xv = F(v)v 1 dv = ln |x| + C. F(v) vv 1 = . F(v) v x 118. x2 y = y2 + xy,y(1) = 1. 119. x2 y = y2 + xy,y(1) = 1.y = (y/x)2 + y/x. v = y/x 120. x2 y = y2 + xy,y(1) = 1.y = (y/x)2 + y/x. v = y/x xv = v2 + v v = v2 , 121. x2 y = y2 + xy,y(1) = 1.y = (y/x)2 + y/x. v = y/x xv = v2 + v v = v2 , 1 dv = ln |x| + C, v2 1 = ln |x| + C, v 1 v= . ln |x| + C 122. () 1 , ln |x| + C x y= . ln |x| + Cy/x = 123. () 1 , ln |x| + C x y= . ln |x| + Cy/x =1 = y(1) =1 1 = . ln |1| + C C 124. () 1 , ln |x| + C x y= . ln |x| + Cy/x =1 = y(1) = y=1 1 = . ln |1| + C C x . ln |x| 1 125. () 1 , ln |x| + C x y= . ln |x| + Cy/x = 126. () 1 , ln |x| + C x y= . ln |x| + Cy/x =1 = y(1) =1 1 = . ln |1| + C C 127. () 1 , ln |x| + C x y= . ln |x| + Cy/x =1 = y(1) = y=1 1 = . ln |1| + C C x . ln |x| 1 128. dx = f (x). dt 129. dx = f (x). dt dx = k(x A). dt () dP = kP. dt 130. dx = f (x). dt dx = k(x A). dt () dP = kP. dt x f (x) = 0 . . 131. 010203040506060006000500050004000400030003000200020001000100000 010203040 : .5060 132. 05101502051015205105010105100500-5-5-10 -5-10 05101520-5 051015 : x = 0.3(x 5) x = 0.1x(5 x).20 133. ( ) x t. 134. - dx = kx(M x), dt M. 135. - dx = kx(M x), dt M. 5 lim x(t) = 0 t x(0) > 0, x(0) = 0, x(0) < 0. 136. y=5 y=0 137. y=5 y=0 138. 139. h . x () t (). M . k > 0 . 140. h . x () t (). M . k > 0 . dx = kx(M x) h. dt 141. h . x () t (). M . k > 0 . dx = kx(M x) h. dt : A, B =kM (kM)2 4hk . 2k 142. (h 1, 1.6) 051015201005101520101010888855552222000005101520051015 : x = 0.1x(8 x) 1 x = 0.1x(8 x) 1.6.20 143. (h 2) 0510152010108855220005101520 : x = 0.1x(8 x) 2.