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8 jednacine prvog reda

8. Dokazati da je y = eC tan(x/2) opste resenje diferencijalnejednacine

ydx+ (1 + x2)dy = 0,

a zatim odrediti partikularno resenje za koje je: (1) y(π/2) = e,(2) y(π/2) = 1.

9. Data je diferencijalna jednacina

y′ = 3√

(y − x)2 + 1.

(1) Dokazati da je y =

(

x− C3

)3

+ x njeno opste resenje.

(2) Odrediti partikularno resenje za koje je y(1) = 2.(3) Da li postoji resenje date jednacine koje se ne moze dobiti

iz opsteg resenja ?

10. Dokazati da je y = Cx + C2 opste resenje, a y = −x2/4singularno resenje diferencijalne jednacine

xy′ + y′2 − y = 0.

11. Dokazati da je x2 = C(y − C) opste resenje, a y = 2x iy = −2x singularna resenja diferencijalne jednacine

xy′2 − 2yy′ + 4x = 0.

12. Dokazati da je y = Cex + 1/C opste resenje, a y = 2ex/2 iy = −2ex/2 singularna resenja diferencijalne jednacine

y′2 − yy′ + ex = 0.

13. Dokazati da je y = Cx+C lnC opste resenje, a y = e−(x+1)

singularno resenje diferencijalne jednacine

y = xy′ + y′ ln y′.

Jednacine sa razdvojenim promenljivim

U zadacima 14.-21. resiti datu diferencijalnu jednacinu.

14. x(y + 1)dx− y(x2 + 1)dy = 0.

jednacine prvog reda 9

15. y(1− x2)dy − x(1− y2)dx = 0.

16. xydx+ (1 + y2)√1 + x2dy = 0.

17. y′ + y2 = 1.

18.√

1− y2dx+√1− x2dy = 0.

19. (xy2 + y2)dx+ (x2 − x2y)dy = 0.

20. ex tan ydx = (ex − 1) cos−2 ydy.

21. 2y′ = cos(x− y)− cos(x+ y).

U zadacima 22.-30. odrediti opste resenje date diferencijalnejednacine.

22. y′ = xy + x+ y + 1.

23. x+ xy + y′(y + xy) = 0.

24. (xy2 + x)dx+ (y − x2y)dy = 0.

25. y′ tan x− y = 1.

26.√

1− y2dx+ y√1− x2dy = 0.

27. x√

1 + y2 + yy′√1 + x2 = 0.

28. e−y(1 + y′) = 1.

29. ey(1 + x2)dy − 2x(1 + ey)dx = 0.

30. ex sin3 y + (1 + e2x) cos yy′ = 0.

U zadacima 31.-41. odrediti partikularno resenje date diferen-cijalne jednacine koje zadovoljava dati uslov.

31. y − xy′ = 2(1 + x2y′), y(1) = 1.


32. y′ sin x− y cos x = 0, y(π/2) = 1.

33. y′ + sin(x− y) = sin(x+ y), y(π) = π/2.

34. ydx+ (1 + x2)dy = 0, y(1) = 1.

35. x2(y3 + 5)dx+ (x3 + 5)y2dy = 0, y(0) = 1.

36. x√

1 + y2dx+ y√1 + x2dy = 0, y(

√3) = 0.

37. tan ydx− x ln xdy = 0, x(π/2) = e.

38. (1 + y2)dx+ (1 + x2)dy = 0, y(1) = 2.

39. (1 + ex)yy′ = ex, y(0) = 1.

40. y′ sin x = y ln y, y(π/2) = e.

41. x3y′ sin y = 2, y → π/2 (x→∞).

U zadacima 42.-48., najpre odgovarajucom smenom svesti datudiferencijalnu jednacinu na jednacinu sa razdvojenim promenljivim,a zatim odrediti njeno opste resenje.

42. (x+ y)y′ = 1.

43. (3x+ y)y′ = 1.

44. y′ =√2x+ y − 3.

45. y′ = cos(x+ y).

46. (x+ y)dx+ (x+ y − 1)dy = 0.

47. (x+ y + 1)dx+ (2x+ 2y − 1)dy = 0.

48. (x− y + 2)dx+ (x− y + 3)dy = 0.


49. Odrediti jednacinu familije krivih kod kojih tacka dodiratangente i krive deli odsecak tangente izme -du koordinatnih osa nadva podudarna dela.

50. Odrediti jednacinu krive koja sadrzi tacku A(2, 0) i za kojuje odsecak tangente izme -du dodirne tacke i ordinatne ose jednak 2.

Homogene jednacine


51. xdy = (x+ y)dx.

52. 2x2dy = (x2 + y2)dx.

53. (x+ y)dx+ (x− y)dy = 0.

54. (x2 + y2)dx− 2xydy = 0.


55. y − xy′ = y lnx

y.

56. xy′ =√

x2 − y2 + y.

57. y − xy′ = x+ yy′.

58. xy′ cosy

x= y cos

y

x− x.

59. y′ +x2 + y2

xy= 0.

60. xydy − y2dx = (x+ y)2e−y/xdx.

61. (x2 − xy)dy + y2dx = 0.


62. y′ = ey/x +y

x+ 1.

63. xy′ = x siny

x+ y.

64. xy′ + x cosy

x− y + x = 0.

65. xy′ − y =√

x2 + y2.

66. xy′ = y lny

x.

67. x2y′ = xy + y2e−x/y.

68. x cosy

x(ydx+ xdy) = y sin

y

x(xdy − ydx).

U zadacima 69.-71. odrediti partikularno resenje date diferen-cijalne jednacine koje zadovoljava dati uslov.

69. y′ =ex/y

1 +x

yex/y

, y(0) = −1.

70. y′(x2 − y2) = xy, y(0) = e.

71. (xy′ − y) arctan yx= x, y(1) = 0.

72. Odrediti jednacinu krive za koju je odsecak tangente izme -dukoordinatnih osa jednak proizvodu koordinata dodirne tacke.

Jednacine koje se svode na homogene

U zadacima 73.-79. odgovarajucom smenom svesti datu difer-encijalnu jednacinu na homogenu i zatim odrediti njeno opste resenje.

73. (x+ y − 2)dx+ (x− y + 4)dy = 0.


74. (x+ y + 1)dx+ (2x+ 2y − 1)dy = 0.

75. y′ = − x− 2y + 5

2x− 5y + 4.

76. y′ =x+ y − 3

x− y − 1.

77. y′ =2x+ y − 1

4x+ 2y + 5.

78. y′ =

(

y + 1

x+ y − 2

)2

.

79. (3y − 7x+ 7)dx− (3x− 7y − 3)dy = 0.

U zadacima 80.-81. smenom y = zα svesti datu diferencijalnujednacinu na homogenu, a zatim odrediti njeno opste resenje.

80. (x2y2 − 1)dy + 2xy3dx = 0.

81. 2x4yy′ + y4 = 4x6.

Linearne jednacine

U zadacima 82.-93. resiti datu linearnu diferencijalnu jednaci-nu.

82. y′ + 2xy = 2xe−x2.

83. y′ + sin x = y + cos x.

84. y′ + 2xy = 2x2e−x2.

85. xy′ + 2y = x2.

86. (x3 + y)dx− xdy = 0.


87. y′ +1− 2x

x2y = 1.

88. y′ + y = cos x.

89. ex2y′ + 2xyex

2= x sin x.

90. y′ − 2x+ 1

x2 + x+ 1y = cos x− 2x+ 1

x2 + x+ 1sin x.

91. y′ + y tan x =1

cos x.

92. (1 + x2)y′ − 2xy = (1 + x2)2.

93. (x− x3)y′ + (2x2 − 1)y − x3 = 0.

U zadacima 94.-98. odrediti partikularno resenje date linearnediferencijalne jednacine koje zadovoljava dati uslov.

94. (x2 − 1)y′ + 2xy − 4x = 0, y(2) = 2.

95. xy′ − y

x+ 1= x, y(1) = 0.

96. y′ + y cos x = sin x cos x, y(0) = 0.

97. y′ − y tan x =1

cos x, y(0) = 1.

98. x(x− 1)y′ + y = x2(2x− 1), y(2) = 4.

U zadacima 99.-105. resiti datu diferencijalnu jednacinu kojaje linearna po x.

99. (y2 − 6x)y′ + 2y = 0.

100. (x− 2xy − y2)y′ + y2 = 0.

101. y′ =1

2x− y2.


102. (y2 + 1)dx = (xy + y2 + 1)dy.

103. y′ =y

2y ln y + y − x.

104. (1 + y2)dx =(√

1 + y2 sin y − xy)

dy.

105. (1 + y2)dx = (arctan y − x)dy.

106. Odrediti ono resenje diferencijalne jednacine

y′ sin 2x = 2(y + cos x)

koje je ograniceno kad x→ π/2.

107. Odrediti ono resenje diferencijalne jednacine

y′ sin x− y cos x = −sin2 x

x2

koje tezi nuli kad x→∞.

Bernulijeva jednacina

U zadacima 108.-129. resiti datu Bernulijevu diferencijalnujednacinu.

108. y′ − xy = −xy3.

109. y′ + 2xy = 2xy2.

110. y′ + 2xy = 2x3y3.

111. xy′ + y = y2 ln x.

112. y′ +y

x+ 1+ y2 = 0.

113. 3xy2y′ − 2y3 = x3.


114. xy′ − 4y − x2√y = 0.

115. y′(x2y3 + xy) = 1.

116. (y ln x− 2)ydx = xdy.

117. (x3 + ey)y′ = 3x2.

118. y′ + 2xy = y2ex2.

119. y′ − 2yex = 2√yex.

120. 2y′ ln x+y

x=cos x

y.

121. 2y′ sin x+ y cos x = y3 sin2 x.

122. (x2 + y2 + 1)dy + xydx = 0.

123. y′ − y cos x = y2 cos x.

124. dy + (xy − xy3)dx = 0.

125. xy′ − y2 ln x+ y = 0.

126. y′ +xy

1− x2= x√y.

127. y′ +2y

x=

2√y

cos2 x.

128. (1 + x2)y′ − 2xy = 4√

y(1 + x2) arctan x.

129. y′x3 sin y + 2y = xy′.

Jednacine sa totalnim diferencijalom

U zadacima 130.-153. resiti datu diferencijalnu jednacinu satotalnim diferencijalom.

130. (y − 3x2)dx+ (x− 4y)dy = 0.


131. x(2x2 + y2) + y(x2 + 2y2)y′ = 0.

132. (3x2 + 6xy2)dx+ (6x2y + 4y3)dy = 0.

133.

(

x√

x2 + y2+

1

x+

1

y

)

dx =

(

x

y2− y√

x2 + y2+

1

y

)

dy.

134.

(

2x+x2 + y2

x2y

)

dx =x2 + y2

xy2dy.

135.

(

1

x2+

3y2

x4

)

dx =2y

x3dy.

136. xdx+ ydy =ydx− xdyx2 + y2

.

137. eydx+ (xey − 2y)dy = 0.

138. yxy−1dx+ xy ln xdy = 0.

139. (3y2 + 2xy + 2x)dx+ (6xy + x2 + 3)dy = 0.

140. 2x(

1 +√

x2 − y)

dx−√

x2 − ydy = 0.

141.

(

1

x− y2

(x− y)2

)

dx+

(

x2

(x− y)2− 1

y

)

dy = 0.

142.

(

sin 2x

y+ x

)

dx+

(

y − sin2 x

y2

)

dy = 0.

143. (3x2 − 2x− y)dx+ (2y − x+ 3y2)dy = 0.

144.

(

xy√1 + x2

+ 2xy − y

x

)

dx+(√

1 + x2 + x2 − ln x)

dy = 0.

145.xdx+ ydy√

x2 + y2+xdy − ydx

x2= 0.


146.

(

sin y + y sin x+1

x

)

dx+

(

x cos y − cos x+1

y

)

dy = 0.

147. (3x2y + y3)dx+ (x3 + 3xy2)dy = 0.

148. e−ydx+ (1− xe−y)dy = 0.

149.xdx+ (2x+ y)dy

(x+ y)2= 0.

150.

(

y2

(x− y)2− 1

x

)

dx+

(

1

y− x2

(x− y)2

)

dy = 0.

151.

(

x

sin y+ 2

)

dx+(x2 + 1) cos y

cos 2y − 1dy = 0.

152. (sin xy + xy cos xy)dx+ x2 cos xydy = 0.

153.y + sin x cos2 xy

cos2 xydy +

(

x

cos2 xy+ sin y

)

dy = 0.

U zadacima 154.-157. odrediti partikularno resenje date difer-encijalne jednacine sa totalnim diferencijalom koje zadovoljava datiuslov.

154.2x

y3dx+

y2 − 3x2

y4dy = 0, y(1) = 1.

155.(x+ 2)dx+ ydy

(x+ y)2= 0, y(1) = 0.

156. 3x2eydx+ (x3ey − 1)dy = 0, y(0) = 1.

157. 2x cos2 ydx+ (2y − x2 sin 2y)dy = 0, y(0) = 0.

Integracioni faktor

U zadacima 158.-166. odrediti integracioni faktor oblika µ(x)i opste resenje date diferencijalne jednacine.


158. (x2 − y)dx+ xdy = 0.

159. (1− x2y)dx+ x2(y − x)dy = 0.

160.(

1− xy

)

dx+

(

2xy +x

y+x2

y2

)

dy = 0.

161. (x2 − 3y2)dx+ 2xydy = 0.

162. (x4 ln x− 2xy3)dx+ 3x2y2dy = 0.

163.(

e2x − y2)

dx+ ydy = 0.

164. (x sin y + y)dx+ (x2 cos y + x ln x)dy = 0.

165. (x+ sin x+ sin y)dx+ cos ydy = 0.

166. (2x2y + 2y + 5)dx+ (2x3 + 2x)dy = 0.

U zadacima 167.-172. odrediti integracioni faktor oblika µ(y)i opste resenje date diferencijalne jednacine.

167. y2dx+ (yx− 1)dy = 0.

168. (2xy2 − 3y3)dx+ (7− 3xy2)dy = 0.

169. 2xy ln ydx+(

x2 + y2√

y2 + 1)

dy = 0.

170. (sin x+ ey)dx+ cos xdy = 0.

171. (3x2 cos y − sin y) cos ydx− xdy = 0.

172. (1 + 3x2 sin y)dx− x cot ydy = 0.

U zadacima 173.-176. odrediti integracioni faktor koji je oblikaµ(x, y) i opste resenje date diferencijalne jednacine.

173. (3y2 − x)dx+ (2y3 − 6xy)dy = 0.


174. (x2 + y2 + 1)dx− 2xydy = 0.

175. xdx+ ydy + x(xdy − ydx) = 0.

176. ydx− (x+ x2 + y2)dy = 0.


177. y(1 + xy)dx− xdy = 0.

178.y

xdx+ (y3 − ln x)dy = 0.

179. (2x2y + 2y + 5)dx+ (2x3 + 2x)dy = 0.

180. (x4 ln x− 2xy3)dx+ 3x2y2dy = 0.

181. (x cos y − y sin y)dy + (x sin y + y cos y)dx = 0.

Razne jednacine


182. x2dy + (3− 2xy)dx = 0.

183. x(x2 + 1)y′ + y = x(1 + x2)2.

184.xy′ − yx

= tany

x.

185. y′ =1 + y2

xy(1 + x2).

186. y′ = e2x − exy.

187. y′ =1

x cos y + sin 2y.


188. y′ = (x+ y)2.

189. y′ =4x− y + 3

y − x+ 1.

190. y′ =2(y + 2)2

(x+ y − 1)2.

191. y′ =y2 − x

2y(x+ 1).

192. y′ =y3

2(xy2 − x2).

193. yy′ + x =1

2

(

x2 + y2

x

)2

.

194. y′ =y2

2xy + 3.

195. e2x − ex+y(1 + y′) = 0.

196. y′ =y

x(1 + ln y − ln x).

197. (x+ y)dx+ (x− y)dy = 0.

198. y′ + xy2 +y

x= 0.

199. (x+ ln y)dx+

(

1 +x

y+ sin y

)

dy = 0.

200. y′ + sinx+ y

2= sin

x− y2

.


Resenja:

6. y = eπ/4−arctan x.

7. y = 0.

8. (1) y = etan(x/2), (2) y = 1.

9. (2) y = ((x+ 2)/3)3 + x, (3) y = x.

14. Cey = (y + 1)√x2 + 1, y = −1 je s.r.

15. 1− y2 = C(1− x2), (C 6= 0), y = ±1, x = ±1 su s.r.

16.√1 + x2 + ln |y|+ y2/2 = C, y = 0 je s.r.

17. y = tanh(x+ C), y = coth(x+ C), y = ±1 su s.r.

18. x√

1− y2 + y√1− x2 = C, y = ±1 su s.r.

19. ln(x/y) = (x+ y)/(xy) + C, x = 0 i y = 0 su s.r.

20. tan y = (ex − 1)C, x = 0 je s.r.

21. y = 2 arctan(Ce− cos x) + nπ, (C 6= 0), y = kπ.

22. Ce(x+1)2/2 − 1.

23. x+ y = ln |C(x+ 1)(y + 1)|.24. 1 + y2 = C(1− x2).25. y = C sin x− 1.

26.√

1− y2 = arcsin x+ C.

27.√1 + x2 +

√

1 + y2 = C.

28. ex = C(1− e−y).29. 1 + ey = C(1 + x2).

30. arctan ex = 1/(2 sin2 y) + C.

31. y = (2 + x)/(1 + 2x).

32. y = sin x.

33. tan(y/2) = e2 sin x.

34. y = eπ/4−arctan x.

35. (x3 + 5)(y3 + 5) = 30.

36.√1 + x2 +

√

1 + y2 = 3.

37. x = esin y.


38. (x+ y)/(1− xy) = −3.39. y2 = 2 ln(1 + ex) + 1− 2 ln 2.

40. y = etan(x/2).

41. y = arccos(1/x2).

42. y − ln |x+ y + 1| = C.

43. (3x+ y)/3− ln |9x+ 3y + 1|/9 = x+ C.

44. 2√2x+ y − 3− 4 ln(

√2x+ y − 3 + 2) = x+ C.

45. y = −x+ 2 arctan(x+ C) + 2nπ, y = −x+ (2n+ 1)π.

46. (x+ y − 1)2 + 2x = C.

47. x+ 2y + 3 ln |x+ y − 2| = C.

48. ln |2x− 2y + 5| − 2(x+ y − 2) = C.

49. y = C/x.

50. y =√4− x2 + 2 ln |(2−

√4− x2)/x|.

51. y = x(ln |x|+ C), x = 0 je s.r.

52. x = Ce−2x/(y−x), x = 0 i y = x su s.r.

53. x2 + 2xy − y2 = C.

54. x2 − y2 − Cx = 0.

55. y = xeCx.

56. y = x sin lnCx, (e−π/2 ≤ Cx ≤ eπ/2).

57. arctany

x+ ln(C

√

x2 + y2) = 0, C > 0).

58. ln |x|+ siny

x= C.

59. x2(x2 + 2y2) = C.

60. (x+ y) lnCx = xey/x.

61. y = Cey/x.

62. ey/x =Cx

1− Cx.

63. y = 2x arctanCx.

64. tany

2x= ln(C/x).

65. x2 = C2 + 2Cy.


66. y = xe1+Cx.

67. ex/y + ln |x| = C.

68. xy cos(y/x) = C.

69. x− y ln(ln |y| − 1) = 0.

70. (x/y)2 + ln y2 = 2.

71.√

x2 + y2 = e(y/x) arctan(y/x).

72. x = Ce2√y/x ili x = Ce−2

√y/x.

73. x2 + 2xy − y2 − 4x+ 8y = C.

74. x+ 2y + 3 ln |x+ y − 2| = C.

75. y − x− 3 = C(y + x+ 1)3.

76. C√

(x− 2)2 + (y − 1)2 = earctan(y+1/x−3).

77. 10y − 5x+ 7 ln |10x+ 5y + 9| = C.

78. y + 1 = Ce−2 arctan(y+1/x−3).

79. (x+ y − 1)5(x− y − 1)2 = C.

80. 1 + x2y2 = Cy, (α = −1).81. α = 3/2.

82. y = (x2 + C)e−x2.

83. y = Cex + sin x.

84. y = (2x3/3 + C)e−x2.

85. y = x2/4 + C/x2.

86. y = x3/2 + Cx.

87. y = Cx2e1/x + x2.

88. y = Ce−x + (cos x+ sin x)/2.

89. y = Ce−x2+ (sin x− x cos x)e−x2 .

90. y = C(x2 + x+ 1) + sin x.

91. y = C cos x+ sin x.

92. y = (x+ C)(1 + x2).

93. y = x+ Cx√1− x2.

94. y = 2.


95. y = x(x− 1 + ln |x|)/(x+ 1).

96. y = sin x− 1 + e− sin x.

97. y = x/ cos x+ 1.

98. y = x2.

99. x = y2/2 + Cy3.

100. x = y2(1 + Ce1/y).

101. x = Ce2y + y2/2 + y/2 + 1/4.

102. x/√

y2 + 1− ln |y +√

y2 + 1| = C.

103. x = y ln y + C/y.

104. x√

1 + y2 + cos y = C.

105. x = arctan y − 1 + Ce− arctan y.

106. y = tan x− 1/ cos x.

107. y = sin x/x.

108. y2(1 + Ce−x2) = 1.

109. y = 1/(1 + Cex2).

110. 1/y2 = Ce2x2+ x2 + 1/2.

111. y = 1/(1 + Cx+ ln x).

112. y = 1/((1 + x)(C + ln(1 + x))).

113. y3 = x3 + Cx.

114. y = x4 ln2 |Cx|/4.115. y(Cx+ ln x+ 1) = 1.

116. y(Cx2 + ln x2 + 1) = 4.

117. x3e−y = C + y.

118. y = e−x2/(C − x).

119.√y + 1 = Cee

x

.

120. y2 ln x = C + sin x.

121. y2(C − x) sin x = 1.

122. y4 + 2x2y2 + 2y2 = C.

123. y = 1/(Ce− sin x − 1).


124. y2(Cex2+ 1) = 1.

125. 1/y = 1 + ln x+ Cx.

126.√y = C 4

√1− x2 − (1− x2)/3.

127. y = ((C + ln | cos x|)/x+ tan x)2.

128. y = 0 ili y = (1 + x2)(arctan2 x+ C)2.

129. y = 0 ili y = (C − cos y)x2.130. 2y2 − xy + x3 = C.

131. x4 + x2y2 + y4 = C.

132. x3 + 3x2y2 + y4 = C.

133.√

x2 + y2 + ln |xy|+ x/y = C.

134. x3y + x2 − y2 = Cxy.

135. x+ y2 = Cx3.

136. x2 + y2 − 2 arctan(x/y) = C.

137. xey − y2 = C.

138. xy = C.

139. 3xy2 + x2y + 3y + x2 = C.

140. x2 + 2(x2 − y)3/2/3 = C.

141. xy/(x− y) + ln(x/y) = C.

142. sin2 x/y + (x2 + y2)/2 = C.

143. x3 + y3 − x2 − xy + y2 = C.

144. y√1 + x2 + x2y − y ln x = C.

145.√

x2 + y2 + y/x = C.

146. x sin y − y cos x+ ln |xy| = C.

147. xy(x2 + y2) = C.

148. y + xe−y = C.

149. ln(x+ y)− x/(x+ y) = C.

150. ln(y/x)− xy/(x− y) = C.

151. x2 + 1 = 2(C − 2x) sin y.

152. x sin xy = C.

153. tan(xy)− cos x− cos y = C.


154. y = x.

155. ln |x+ y| = y/(x+ y).

156. x3ey − y = −1.157. 2y2 + x2 cos y + x2 = 0.

158. µ = 1/x2, x+ y/x = C.

159. µ = 1/x2, xy2 − 2x2y − 2 = Cx.

160. µ = 1/x, ln |x|+ ln |y|+ y2 − xy = C.

161. µ = 1/x4, y2 = Cx3 + x2.

162. µ = 1/x4, y3 + x3(ln x− 1) = Cx2.

163. µ = e−2x, y2 = (C − 2x)e2x.

164. µ = 1/x, x sin y + y ln x = C.

165. µ = ex, 2ex sin y + 2ex(x− 1) + ex(sin x− cos x) = C.

166. µ = 1/(1 + x2), 5 arctan x+ 2xy = C.

167. µ = 1/y, xy − ln y = 0.

168. µ = 1/y2, x2 − 7/y − 3xy = C.

169. µ = 1/y, x2 ln y + (y2 + 1)3/2/3 = C.

170. µ = e−y, e−y cos x = C + x.

171. µ = 1/ cos2 y, x3 − x tan y = C.

172. µ = 1/ sin y, x/ sin y + x3 = C.

173. µ = 1/(x+ y2)3, (x+ y2)2C = x− y2.174. µ = 1/(1 + y2 − x2)2, 1 + y2 − x2 = Cx.

175. µ = 1/(x2 + y2)3/2, y − 1 = C√

x2 + y2.

176. µ = 1/(x2 + y2), arctan(x/y)− y = C.

177. x2 + 2x/y = C.

178. y2/2 + ln x/y = C.

179. 5 arctan x+ 2xy = C.

180. y3 + x3(ln x− 1) = Cx2.

181. (x sin y + y cos y − sin y)ex = C.

182. y = Cx2 + 1/x.

183. y = C√x2 + 1/x+ (1 + x2)2/(3x).


184. sin(y/x) = Cx.

185. (1 + x2)(1 + y2) = Cx2.

186. y = Ce−ex

+ ex − 1.

187. x = Cesin y − 2(1 + sin y).

188. arctan(x+ y) = x+ C.

189. (2x− y + 1/3)(2x+ y + 5)3 = C.

190. e−2 arctan((y+2)/(x−3)) = C(y + 2).

191. y2 = x+ (x+ 1) ln(C/(x+ 1)).

192. y2e−y2/x = C.

193. Cx = 1− x/(x2 + y2).

194. x = Cy2 − 1/y.

195. e2x/2− ex+y − ey = C.

196. y/x = eCx.

197. x2 − y2 + 2xy = C.

198. y(x2 + Cx) = 1.

199. x2/2 + x ln y + y cos y = C.

200. ln | tan(y/4)| = C − 2 sin(x/2).

JEDNACINE KOJIMA SE MOZE SNIZITI RED

F (x, y(k), y(k+1), . . . , y(n)) = 0


1. y′′′ = x+ cos x.

2. y(4) = x.

3. xy′′ + y′ − x2 = 0.

4. x3y′′ + x2y′ = 1.

5. y′′x ln x = y′.

6. xy′′ − y′ = exx2.

7. y(4) =√y′′′.

8. xy′′ = y′.

9. (1 + x2)y′′ + y′2+ 1 = 0.

10. (1 + x2)y′′ + 2xy′ = x3.

11. xy′′ = y′ lny′

x.

12. y′′′ = y′′3.

30 jednacine kojima se moze sniziti red

13. xy′′′ − y′′ = 0.

14. y′′x ln x = y′.

15. xy′′ = (1 + 2x2)y′.

16. y′′ + y′ tan x = sin 2x.

17. y′′ tan x = y′ + 1.

U zadacima 18.-24. odrediti partikularno resenje date diferen-cijalne jednacine koje zadovoljava date uslove.

18. y′′(x+ 2)5 = 1, y(−1) = 1/12, y′(−1) = −1/4.

19. y′′ = xex, y(0) = y′(0) = 0.

20. y′′ + y′ + 2 = 0, y(0) = 0, y′(0) = −2.

21. y′′ = y′ ln y′, y(0) = 0, y′(0) = 1.

22. y′′ = 4 cos 2x, y(0) = y′(0) = 0.

23. y′′ =1

cos2 x, y

(π

4

)

=ln 2

2, y′

(π

4

)

= 1.

24. y′′′ =6

x3, y(1) = 2, y′(1) = y′′(1) = 1.

F (y, y′, y′′, . . . , y(n)) = 0


25. y′2+ yy′′ = 0.

26. yy′′ = y′2 − y′3.

27. yy′′ + y′2= 0.

jednacine kojima se moze sniziti red 31

28. y′′ + y′2= 2e−y.

29. y′′ + 2yy′3= 0.

30. yy′′ + y′2= y2 ln y.

31. y′′ tan y = 2y′2.

32. yy′′ = y2y′ + y′2.

33. y′′y3 = −1.

34. y′′ = 1 + y′2.

35. 2yy′′ = y′2.

36. yy′′ = y′2.

37. 2yy′′ = 1 + y′2.

38. yy′′ = y′ + y′2.

39. 2yy′′ = 1 + y′2.

40. 2yy′′ − 3y′2= 4y2.


41. y′′ = 2yy′, y(0) = y′(0) = 1.

42. 3y′y′′ = 2y, y(0) = y′(0) = 1.

43. 2y′′ = 3y2, y(−2) = 1, y′(−2) = −1.

44. y′′ = e2y, y(0) = 0, y′(0) = 1.


F (x, y, y′, . . . , y(n)) = 0, F - homogena funkcija


45. x2yy′′ = (y − xy′)2.

46. xyy′′ − xy′2 = yy′.

47. xyy′′ + xy′2= yy′.

48. xyy′′ + xy′2= 3yy′.

49. y′′ +y′

x+

y

x2=y′

2

y.

50. x3y′′ = (y − xy′)(y − xy′ − x).

Resenja:

1. y = x4/24− sin x+ C1x2 + C2x+ C3.

2. y = x5/120 + C1x3 + C2x

2 + C3x+ C4.

3. y = C1 ln |x|+ x3/9 + C2.

4. y = 1/x+ C1 ln x+ C2.

5. y = C1x(ln x− 1) + C2

6. y = ex(x− 1) + C1x2 + C2

7. y = 2(x/2 + C1)5/15 + C2x

2/2 + C3x + C4 i y = D1x2/2 +

D2x+D3.

8. y = C1x2 + C2.

9. y = (1 + C21 ) ln(x+ C1)− C1x+ C2.

10. y = x3/12− x/4 + C1 arctan x+ C2.

11. y = (C1x+ C21 )e

x/C1+1 + C2.

12. y = (C1 − 2x)3/2/3 + C2x+ C3.

13. y = C1x3 + C2x+ C3.

jednacine kojima se moze sniziti red 33

14. y = C1x(ln x− 1) + C2.

15. y = C1ex2 + C2.

16. y = C1 sin x− x− sin 2x/2 + C2.

17. y = C2 − C1 cos x− x.18. y = 1/(12(x+ 2)3).

19. y = (x− 2)ex + x+ 2.

20. y = −2x.21. y = x.

22. y = 1− cos 2x.23. y = − ln(cos x).24. y = 3 ln x+ 2x2 − 6x+ 6.

25. y2 = 2C1x+ C2.

26. ln |y|+ C1y = C1x+ C2 i y = x+ C.

27. y2 = C1x+ C2.

28. ey + C1 = (x+ C2)2.

29. y3 + C1y + C2 = 3x.

30. ln y = C1ex + C2e

−x.

31. cot y = C2 − C1x.

32. y = C1C2eC1x/(1− C2e

C1x) i y = C 6= 0.

33. C1y2 + 1 = (C1x+ C2)

2.

34. y = C2 − ln | cos(C1 + x)|.35. y = (C1x+ C2)

2.

36. y = C2eC1x.

37. 4(C1y − 1) = (C1x+ C2)2.

38. y = (1 + C2eC1x)/C1.

39. y = (1 + (C1x+ C2)2/4)/C1.

40. y cos(x+ C1) = C2.

41. y = 1/(1− x).42. y = (1 + x/3)3.

43. y = 4/(x+ 4)2.


44. y = − ln |x− 1|.45. y = C2xe

−C1/x.

46. y = C1eC2x

2.

47. y2 = C1x2 + C2.

48. y2 = C1x4 + C2.

49. y = C2|x|C1−ln |x|/2.

50. y = −x ln(C2 lnC1x) i y = Cx.

LINEARNE HOMOGENE JEDNACINE

SA KONSTANTNIM KOEFICIJENTIMA

Koreni realni i jednostruki


1. y′′ − y = 0.

2. y′′ − 9y = 0.

3. y′′ − y′ = 0.

4. 4y′′ − y = 0.

5. y′′ + 4y′ = 0.

6. y′′ + 6y′ + 8y = 0.

7. y′′ + 3y′ − 4y = 0.

8. y′′ − 7y′ + 12y = 0.

9. y′′ + 3y′ + 2y = 0.

10. y′′ − 4y′ + 3y = 0.

11. y′′ + y′ − 2y = 0.

12. 3y′′ − 2y′ − 8y = 0.

36 linearne homogene jednacine

13. y′′′ + 2y′′ − 3y′ = 0.

14. y′′′ − 13y′ − 12y = 0.

15. y′′′ − 13y′′ + 12y′ = 0.

16. y′′′ − 2y′′ − y′ + 2y = 0.

17. y(4) − 5y′′ + 4y = 0.


18. y′′ + 4y′ = 0, y(0) = 7, y′(0) = 8.

19. y′′ − 5y′ + 4y = 0, y(0) = y′(0) = 1.

20. y′′ − 4y′ + 3y = 0, y(0) = 6, y′(0) = 10.

21. y′′′ − y′ = 0, y(0) = 0, y′(0) = −1, y′′(0) = 1.

22. y′′ − y′ − 6y = 0, y(0) = 1, y′(0) = 4.

Me -du realnim korenima ima visestrukih


23. y′′ + 2y′ + y = 0.

24. 4y′′ + 4y′ + y = 0.

25. 9y′′ − 6y′ + y = 0.

26. y′′ + 8y′ + 16y = 0.

27. y′′ − 4y′ + 4y = 0.

28. y′′ + 2ay′ + a2y = 0, a ∈ R.

linearne homogene jednacine 37

29. y′′′ + 2y′′ + y′ = 0.

30. y′′′ − 3y′′ + 4y = 0.

31. y′′′ − y′′ − y′ + y = 0.

32. y′′′ − 5y′′ + 8y′ − 4y = 0.

33. y′′′ + 3y′′ + 3y′ + y = 0.

34. y′′′ − 9y′′ + 15y′ + 25y = 0.

35. y′′′ + 3ay′′ + 3a2y′ + a3y = 0, a ∈ R.

36. y(4) + 2y′′′ + y′′ = 0.

37. y(4) − 2y′′′ + y = 0.

38. y(4) − 8y′′ + 16y = 0.

39. y(4) − 6y′′ + 9y = 0.

40. y(4) + 2y′′′ − 2y′ − y = 0.

41. y(4) + 4y′′′ + 3y′′ − 4y′ − 4y = 0.

42. y(4) − 2y′′′ + 2y′′ − 8y′ + 16y = 0.


43. y′′ − 2y′ + y = 0, y(2) = 1, y′(2) = −2.

44. y′′ − 4y′ + 4y = 0, y(0) = 3, y′(0) = −1.

45. y′′ − 6y′ + 9y = 0, y(0) = 0, y′(0) = 2.

46. 9y′′ − 24y′ + 16y = 0, y(0) = y′(0) = 1.

LINEARNE NEHOMOGENE JEDNACINE

SA KONSTANTNIM KOEFICIJENTIMA

f(x) = Pn(x)

U zadacima 1.-12. metodom neodre -denih koeficijenate resitidatu diferencijalnu jednacinu.

1. y′′ + 2y′ + y = −2.

2. y′′ + 2y′ + 2y = x.

3. y′′ + 2y′ + 2y = 1 + x.

4. y′′ + y′ − 2y = 6x2.

5. y′′ − 2y′ = x2 − x.

6. 2y′′ + 7y′ + 3y = 9x2 + 42x.

7. 4y′′ − y = x3.

8. y′′ + 4y′ + 5y = 5x2 − 32x+ 5.

9. y′′ − 4y = 8x3.

10. y′′′ − y′′ = 12x2 + 6x.

11. y′′′ − y′′ + y′ − y = x2 + x.

12. y(4) + y′′ = x2 + x.

linearne nehomogene jednacine 43

f(x) = eaxPn(x)


13. y′′ − 9y = −6e3x.

14. y′′ + 8y′ + 20y = 29ex.

15. y′′ + 4y′ + 5y = 81xe−2x.

16. y′′ + y′ = 4x2ex.

17. y′′ − 3y′ + 2y = (x2 + x)e3x.

18. y′′ − 6y′ + 13y = ex(x2 − 5x+ 2).

19. y′′′ − 2y′′ + y′ = ex.

20. y′′′ − 3y′′ + 3y′ − y = ex.

21. y(4) − 2y′′′ − y′′ = ex.

f(x) = eax [Pn(x) cos bx+Qm(x) sin bx]


22. y′′ + y = 4x cos x.

23. y′′ + 3y′ + 2y = x sin x.

24. y′′ + y = x2 sin x.

25. y′′ − y′ = ex sin x.

26. y′′ + 2y′ + 5y = e−x sin 2x.

27. y′′ + 3y′ − 4y = e−4x sin x.

44 linearne nehomogene jednacine

28. y′′ + 2y′ + y = e−xx2 cos x.

29. y′′ + 3y′ + 2y = 4 sin 3x+ 2 cos 3x.

30. y′′ − 2y′ + 2y = ex(2 cos x− 4x sin x).

Princip superpozicije


31. y′′ − y′ − 2y = 4x− 2e−x.

32. y′′ − 3y′ + 2y = ex + 6e−x.

33. y′′ − 3y′ = 18x− 10 cos x.

34. y′′ − 2y′ + y = 2 + ex sin x.

35. y′′ − 2y′ + y = sin x+ e−x.

36. y′′ − 2y′ + y = e−x sin x+ 4ex.

37. 2y′′ + 7y′ + 3y = 9x2 + 42x− 50 sin x.

38. y′′ − y′ = 15x2 − 32 + 30 sin 3x+ 12e3x.

39. y′′ + 2y′ − 3y = 2xe−3x + (x+ 1)ex.

40. y′′ + y′ = x2 − e−x + ex.

41. y′′ − 4y′ + 5y = 1 + 8 cos x+ e2x.

42. y′′ + y′ + y + 1 = sin x+ x+ x2.

43. y′′ − 3y′ = 1 + ex + cos x+ sin x.

44. y′′′ − 6y′′ + 12y′ − 8y = 1 + 2e2x.


45. y′′′ − y′′ − y′ + y = x2 − e2x.

46. y′′′ − 3y′′ + 4y′ − 2y = ex + cos x.

47. y′′′ − y′′ + y′ − y = cos x+ 2ex.

48. y′′′ − y′′ + 4y′ − 4y = 3e2x − 4 sin 2x.

49. y′′′ − 4y′ = xe2x + sin x+ x2.

Metoda varijacije konstanti

U zadacima 50.-64. metodom varijacije konstante resiti datudiferencijalnu jednacinu.

50. y′′ − 2y′ + y =ex

x.

51. y′′ − 6y′ + 9y =e3x

x2.

52. y′′ − y′ = 1

ex + 1.

53. y′′ + y =1

cos3 x.

54. y′′ + y =2

sin3 x.

55. y′′ + 2y′ + 2y =1

ex sin x.

56. y′′′ + y′′ =x− 1

x2.

57. y′′ + y =1

cos x.

58. y′′ + 4y =1

sin2 x.


59. y′′ + y =1

cos 2x√cos 2x

.

60. y′′ − y′ = e2x cos ex.

61. y′′ + 4y′ + 4y = e−2x ln x.

62. y′′ + y = ln cos x.

63. y′′ − 2y′ + y =ex√4− x2

.

64. y′′ − 5y′ + 6y =6x2 + 17x+ 13

(x+ 1)3.

Resenja:

1. y = (C1 + C2x)e−x − 2.

2. y = (C1 cos x+ C2 sin x)e−x + (x− 1)/2.

3. y = (C1 cos x+ C2 sin x)e−x + x/2.

4. y = C1ex + C2e

−2x − 3(x2 + x+ 1/2).

5. y = C1 + C2e2x − x3/6.

6. y = C1e−x/2 + C2e

−x + 3x2 − 4.

7. y = C1ex/2 + C2e

−x/2 − x3 − 24x.

8. y = (C1 cos x+ C2 sin x)e−2x + x2 − 8x+ 7.

9. y = C1e2x + C2e

−2x − 2x3 − 3x.

10. y = C1 + C2x+ C3ex − x4 − 5x3 − 15x2.

11. y = C1ex + C2 cos x+ C3 sin x− x2 + 3x− 1.

12. y = C1 + C2x+ C3 cos x+ C4 sin x+ x4/12 + x3/6− x2.13. y = C1e

3x + C2e−3x − xe3x.

14. y = e−4x(cos 2x+ C2 sin 2x) + ex.

15. y = (C1 cos x+ C2 sin x)e−2x + (3x+ 16/9)e−2x.

16. y = C1 + C2e−x + (2x2 − 6x+ 7)ex.


17. y = C1ex + C2e

2x + (x2/2− x+ 1)e3x.

18. y = (C1 cos 2x+ C2 sin 2x)e3x + (x2/8− x/2− 1/32)ex.

19. y = C1 + (C2 + C3x+ x2/2)ex.

20. y = (C1 + C2x+ C3x2 + x3/6)ex.

21. y = C1 + C2x+ (C3 + C4x)ex + x2ex/4.

22. y = C1 cos x+ C2 sin x+ x cos x+ x2 sin x.

23. y = C1e−x + C2e

−2x − (3x/10 − 17/50) cos x + (x/10 +3/25) sin x.

24. y = (C1 + x/4− x3/6) cos x+ (C2 + x2/4) sin x.

25. y = C1 + C2ex − (cos x+ sin x)ex/2.

26. y = (C1 cos 2x+ C2 sin 2x)e−x + xe−x cos 2x/4.

27. y = C1ex + C2e

−4x + (5 sin x/26− cos x/26)e−4x.28. y = (C1 + C2x)e

−x + ((6− x2) cos x+ 4x sin x)e−x.

29. y = C1e−x + C2e

−2x − sin 3x/13− 5 cos 3x/13.

30. y = ex(C1 cos x+ C2 sin x) + x2ex cos x.

31. y = C1e−x + C2e

2x − 2x+ 1 + 2xe−x/3.

32. y = C1ex + C2e

2x + e−x − xex.33. y = C1 + C2e

3x − 3x2 − 2x+ cos x+ 3 sin x.

34. y = 2 + ex(C1 + C2x− sin x).35. y = C1e

x + C2xex + cos x/2 + e−x/4.

36. y = (C1 + C2x+ 2x2)ex + (3 sin x+ 4 cos x)e−x/25.

37. y = C1e−x/2 + C2e

−3x + 3x2 − 4 + 7 cos x− sin x.38. y = C1 +C2e

x − 5x3 − 15x2 +2x+ cos 3x− 3 sin 3x+2e3x.

39. y = C1e−3x + C2e

x − (2x2 + x)e−3x/8 + (2x2 + 3x)ex/16.

40. y = C1 + C2e−x + xe−x + ex/2 + x3/3− x2 + 2x.

41. y = (C1 cos x+ C2 sin x)e2x + cos x− sin x+ e2x + 1/5.

42. y = (C1 cos(√3x/2)+C2 sin(

√3x/2)e−x/2−cos x+x2−x−2.

43. y = C1 + C2e3x + (cos x− 2 sin x)/5− ex/2− x/3.

44. y = (C1 + C2x+ C3x2)e2x + x3e2x/3− 1/8.

45. y = C1ex + C2xe

x + C3e−x + x2 + 2x+ 4− e2x.


46. y = ex(x+C1+C2 cos x+C3 sin x)+ cos x/10+3 sin x/10.

47. y = C1ex + C2 cos x+ C3 sin x− x(cos x+ sin x)/4 + xex.

48. y = C1ex + C2 cos 2x + C3 sin 2x + 3e2x/8 − x cos 2x/5 +

2x sin 2x/5.

49. y = C1+C2e2x+C3e

−2x+ cos x/5−x3/12−x8+e2x(2x2−3x)/32.

50. y = ex(x ln |x|+ C1x+ C2).

51. y = (C1 + C2x)e3x + (ln(1/x)− 1)e3x.

52. y = C1ex + C2 + (ex + 1) ln(1 + e−x).

53. y = C1 cos x+ C2 sin x− cos 2x/(2 cos x).54. y = C1 cos x+ C2 sin x+ cos 2x/ sin x.

55. y = (C1 − x)e−x cos x+ (C2 + ln | sin x|)e−x sin x.56. y = C1 + C2x+ C3e

−x + 1− x+ x ln |x|.57. y = C1 cos x+ C2 sin x+ x sin x+ cos x ln(cos x).

58. y = (C1 − ln | sin x|) cos 2x+ (C2 − x− cot /2) sin 2x.59. y = C ′1 cos x+ C2 sin x−

√cos 2x.

60. y = C1ex + C2 − cos ex.

61. y = (x2 ln x/2− 3x2/4 + C1 + C2x)e−2x.

62. y = C1 cos x+C2 sin x+ln cos x+sin x ln((1+sin x)/ cos x)−1.

63. y = (C1 +√4− x2 + x arcsin(x/2) + C2x)e

x.

64. y = C1e2x + C2e

3x + 1/(x+ 1).

LINEARNI SISTEMI

Homogen sistem - koreni realni i razliciti

U zadacima 1.-6. odrediti opste resenje datog sistema diferen-cijalnih jednacina.

1.

{

x′ =2x− 3y

y′ =− 2x+ 3y.

2.

{

x′ =4x+ y

y′ =− 2x+ y.

3.

{

x′ =2x− yy′ =3x− 2y.

4.

{

x′ =3x+ 2y

y′ =2x.

5.

x′ =x− y + z

y′ =x+ y − zz′ =2x− y.

6.

x′ =x− 2y − zy′ =− x+ y + z

z′ =x− z.

U zadacima 7.-8. odrediti partikularno resenje datog sistemadiferencijalnih jednacina koje zadovoljava date uslove.

62 linearni sistemi

7.

{

x′ =x+ 2y

y′ =9x+ 4y,

x(0) =3

y(0) =0,

8.

x′ =− x+ y + z

y′ =x− y + z

z′ =x+ y + z,

x(0) =1

y(0) =0

z(0) =0.

Homogen sistem - realni visestruki koreni

U zadacima 9.-17. odrediti opste resenje datog sistema difer-encijalnih jednacina.

9.

{

x′ =2x− yy′ =4x− 2y.

10.

{

x′ =x− 2y

y′ =2x− 3y.

11.

{

x′ =4x− yy′ =x+ 2y.

12.

{

x′ =− x− yy′ =x− 3y.

13.

{

x′ =3x− yy′ =4x− y.

14.

x′ =4x− y − zy′ =x+ 2y − zz′ =x− y + 2z.

15.

x′ =y + z

y′ =x+ y

z′ =− x+ z.

linearni sistemi 63

16.

x′ =4x+ 2y − 2z

y′ =x+ 3y − zz′ =3x+ 3y − z.

17.

x′ =2x− zy′ =x− yz′ =3x− y − z.


18.

{

x′ =x− 2y

y′ =2x− 3y,

x(0) =2

y(0) =− 1.

19.

x′ =x− y + z

y′ =x+ y − zz′ =2x− y,

x(0) =0

y(0) =0

z(0) =1.

20.

x′ =2x− y − zy′ =2x− y − 2z

z′ =− x+ y + 2z,

x(0) =1

y(0) =− 1

z(0) =2.

Homogen sistem - koreni kompleksni


21.

{

x′ =− 7x+ y

y′ =− 2x− 5y.

22.

{

x′ =x− 3y

y′ =3x+ y.

23.

{

x′ =− x− 5y

y′ =x+ y.

64 linearni sistemi

24.

{

x′ =x+ y

y′ =− 2x+ 3y.

25.

x′ =2x+ y

y′ =x+ 3y − zz′ =− x+ 2y + 3z.

26.

x′ =2x− y + 2z

y′ =x+ 2z

z′ =− 2x+ y − z.


27.

{

x′ =x+ y

y′ =− 2x+ 3y,

x(π/2) =1

y(0) =0.

28.

x′ =x− y − zy′ =x+ y

z′ =3x+ z,

x(π/4) =eπ/4

y(π/4) =eπ/4

z(π/4) =eπ/4.

Nehomogen sistem

U zadacima 29.-32. odrediti opste resenje datog sistema difer-encijalnih jednacina ako je data jedna od njegovih fundamentalnihmatrica.

29.

x′ =x+y

t+t

2(et + e−t)

y′ =x

t+ y +

t

2(et − e−t),

, F (t) =

(

tet te−t

tet −te−t)

30.

{

x′ =tx+ y + t

y′ =x+ ty + 1,, F (t) =

(

e(t+1)2/2 e(t−1)2/2

e(t+1)2/2 e(t−1)2/2

)

linearni sistemi 65

31.

{

x′ =(2 sin t− cos t)x+ (sin t− cos t)y − cos ty′ =2(cos t− sin t)x+ (2 cos t− sin t)y + 2 cos t,

F (t) =

(

e− cos t −esin t−e− cos t 2esin t.

)

32.

t2x′ =(t+ 1)x+ y − tz + t− t ln tt2y′ =− x+ (t− 1)y + tz − t

(t2 ln t)z′ =− x− y + tz − ln2 t,

F (t) =

1 + ln t t 0−1− ln t 0 tln t 1 1.


33.

{

x′ =− 5x+ 2y + et

y′ =x− 6y + e−2t.

34.

{

x′ =− x+ 2y + 1

y′ =− 2x+ 3y.

35.

{

x′ =4x− 3y + sin t

y′ =2x− y − 2 cos t.

36.

{

x′ =2x+ y + et

y′ =− 2x+ 2t.

37.

{

x′ =2x− yy′ =− 2x+ y + 18t.

38.

{

x′ =x+ 2y + 16tet

y′ =2x− 2y.

66 linearni sistemi

39.

{

x′ =y

y′ =x+ et + e−t.

40.

x′ =− 2x+ 3y + 4z − 3t

y′ =− 6x+ 7y + 6z + 1− 7t

z′ =x− y + z + t.


41.

{

x′ =x− 2y − e2t

y′ =x+ 4y,

x(0) =2

y(0) =0.

42.

{

x′ =y + et

y′ =− x− sin 2t,x(0) =

1

2y(0) =1.

43.

x′ =y − z − t+ 3

y′ =− 2z + 7

z′ =2y − 2t,

x(π) =2

y(π) =π + 2

z(π) =2.

Sistemi koji nisu dati u normalnom obliku


44.

{

4x′ =− y + cos t

4x′ − y′ =− 3x+ sin t.

45.

{

x′′ =y

y′′ =x.

46.

{

x′′ =3x+ 4y

y′′ =− x− y.

linearni sistemi 67

47.

{

x′′ − 2y′ =− 2x

3x′ + y′′ =8y.

48.

{

x′′ + y′ =− x+ et

x′ + y′′ =1.

Resenja:

1. x = 2C1 + C2e5t, y = 3C1 − C2e

5t.

2. x = 2C1e2t + C2e

3t, y = −2C1e2t − C2e

3t.

3. x = C1et + C2e

−t, y = C1et + 3C2e

−t.

4. x = 2C1e4t + C2e

−t, y = C1e4t − 2C2e

−t.

5. x = C1et + C2e

2t + C3e−t, y = C1e

t − 3C3e−t, z = C1e

t +C2e

2t − 5C3e−t.

6. x = C1 + 3C2e2t, y = −2C2e

2t + C3e−t, z = C1 + C2e

2t −2C3e

−t.

7. x = e7t + 2e−2t, y = 3e7t − 3e−2t.

8. x = e−t/3+ e2t/6+ e−2t/2, y = e−t/3+ e2t/6− e−2t/2, z =−e−t/3 + e2t/3.

9. x = 2C1(1 + 2t) + C2, y = 2C1t+ C2.

10. x = e−t(C1 + C22C1t), y = e−t(C2 + 2C1t).

11. x = e3t(C1 − C2 + C1t), y = e3t(C2 + C1t).

12. x = e−2t(C1 − C2 + C1t), y = e−2t(C2 + C1t).

13. x = et(C2 + C1et), y = et(2C2 − C1 + 2C1t).

14. x = C1e2t + (C2 + C3)e

3t, y = C1e2t + C3e

3t, z = C1e2t +

C2e3t.

15. x = C1 + C3et, y = −C1e

t(C2 + C3 + C3t), z = C1 +et(−C2 − C3t).

16. x = e2t(C1 + C2 + C3 + 2C3t), y = e2t(−C1 + C3t), z =e2t(C2 + 3C3t).

17. x = C1+C2t+C3t2, y = C1+C2(t−1)+C3(t

2−2t+2), z =2C1 + C2(2t− 1) + C3(2t

2 − 2t).

68 linearni sistemi

18. x = e−t(2 + 6t), y = e−t(−1 + 6t).

19. x = (2 + t)et − 2e2t, y = tet, z = (1 + t)et − 2e2t.

20. x = et, y = −et, z = 2et.

21. x = e−6t(C1 cos t + C2 sin t), y = e−6t((C1 + C2) cos t +(C2 − C1) sin t).

22. x = et(C1 cos 3t+ C2 sin 3t), y = et(C1 sin 3t− C2 cos 3t).

23. x = (2C2 − C1) cos 2t − (2C1 + C2) sin 2t, y = C1 cos 2t +C2 sin 2t.

24. x = e2t(C1 cos t+C2 sin t), y = e2t((C1 +C2) cos t+ (C2 −C1) sin t).

25. x = C1e2t+e3t(C2 cos t+C3 sin t), y = e3t((C2+C3) cos t+

(C3−C2) sin t), z = C1e2t+ e3t((2C2−C3) cos t+(C2+2C3) sin t).

26. x = C2 cos t+(C2+2C3) sin t, y = 2C1et+C2 cos t+(C2+

2C3) sin t, z = C1et + C3 cos t− (C2 + C3) sin t.

27. x = e2t−π(cos t+ sin t), y = −2e2t−π sin t.28. x = et(sin 2t+cos 2t), y = et(1/2−cos 2t/2+sin 2t/2), z =

et(−1/2− 3 cos 2t/2 + 3 sin 2t/2).

29. x = C1tet + C2te

−t + t2(et + e−t)/2, y = C1tet − C2te

−t +t2(et + e−t)/2.

30. x = C1e(t+1)2/2 + C2e

(t−1)2/2 − 1, y = C1e(t+1)2/2+

+C2e(t−1)2/2.

31. x = C1e− cos t−C2e

sin t+1, y = −C1e− cos t+2C2e

sin t−2.

32. x = C1(1+ln t)+C2t+ln t, y = −C1(1+ln t)+C3t+1, z =C1 ln t+ C2 + C3 + (1 + ln t)/t.

33. x = C1e−4t + C2e

−7t + 7et/40 + e−2t/5, y = C1e−4t/2 −

C2e−7t + et/40 + 3e−2t/10.

34. x = (C1 + 2C2t)et − 3, y = (C1 + C2 + 2C2t)e

t − 2.

35. x = C1et + 3C2e

2t + cos t − 2 sin t, y = C1et + 2C2e

2t +2 cos t− 2 sin t.

36. x = C1et cos t + C2e

t sin t + et + t + 1, y = C1et(− cos t −

sin t) + C2et(cos t− sin t)− 2et − 2t− 1.

37. x = C1e3t+3t2 +2t+C2, y = −C1e

3t+6t2− 2t+2C2− 2.

linearni sistemi 69

38. x = 2C1e2t +C2e

−3t − (12t+ 13)et, y = C1e2t − 2C2e

−3t −(8t+ 6)et.

39. x = C1et + C2e

−t + t sinh t, y = C1et − C2e

−t + sinh t +t cosh t.

40. x = C1et + C2e

2t + C3e3t, y = C1e

t + 3C3e3t + t, z =

C2e2t − C3e

3t.

41. x = 3e2t − e3t − 2te2t, y = −e2t + e3t + te2t.

42. x = C1 cos t+C2 sin t+ sin 2t/3+ et(sin 2t− cos 2t)/2, y =

C2 cos t− C1 sin t+ 2 cos 2t/3 + et(sin 2t+ cos 2t)/2.

43. x = sin 2t+2, y = t+cos 2t+sin 2t, z = 3+sin 2t−cos 2t.44. x = C1e

−t + C2e−3t, y = −C1e

−t + 3C2e−3t + cos t.

45. x = C1et +C2e

−t +C3 cos t+C4 sin t, y = C1et +C2e

−t −C3 cos t− C4 sin t.

46. x = −2et(C1 + C2 + C2t) − 2e−t(C3 − C4 + C4t), y =et(C1 + C2t) + e−t(C3 + C4t).

47. x = 2C1e2t+2C2e

−2t+2C3 cos 2t+2C4 sin 2t, y = 3C1e2t−

3C2e−2t − C3 sin 2t+ C4 cos 2t.

48. x = C1 +C2t+C3t2− t3/6+ et, y = −(C1 +2C3)t− (C2−

1)t2/2− C3t3/3 + t4/24− et + C4.

PRVI KOLOKVIJUM

Primer 1

1. Odrediti resenje jednacine

(1 + x2)y′ − 4√

y(1 + x2) arctan x = 2xy

koje zadovoljava uslov y(0) = 0.

2. Odrediti opste resenje jednacine

y′′ − 2y′ + y = −2x2 + 4ex + 2 sin x.

3. Resiti sistem diferencijalnih jednacina

dx

dt=x2y + tx

dy

dt=xy2 − ty

Resenja

1. Jedno resenje date jednacine je funkcija y : x 7→ 0. Ako je y 6= 0,

jednacina je ekvivalentna Bernulijevoj jednacini

y′ − 2x

1 + x2y =

4√y arctan x√1 + x2

,

koja se smenom z =√y svodi na linearnu jednacinu

z′ − x

1 + x2z =

2 arctan x√1 + x2

Prvi kolokvijum 71

cije je resenje familija funkcija

z : x 7→√1 + x2(arctan2 x + C), C ∈ R.

Prema tome, opste resenje date jednacine je familija funkcija

y : x 7→ (1 + x2)(arctan2 x + C)2, C ∈ R,

a trazeno partikularno resenje je funkcija

y : x 7→ (1 + x2) arctan4 x.

Data jednacina, dakle, ima dva resenja.

2. Neka je L[y] izraz na levoj strani date jednacine. Karakteristicna

jednacina je k2 − 2k + 1 = 0, pa je opste resenje yh odgovarajuce ho-

mogene jednacine, L[y] = 0, dvoparametarska familija funkcija definisana

jednakoscu

yh(x;C,D) = Cex +Dxex.

Partikularno resenje yp date jednacine je zbir yp1 + yp2 + yp3 , gde su yp1 ,

yp2 i yp3 partikularna resenja jednacina

L[y] = −2x2, L[y] = 4ex, L[y] = 2 sin x. (∗)

Kako je

yp1(x) = ax2 + bx + c, yp2(x) = dx2ex, yp3(x) = e cos x + f sin x,

iz jednacina (∗) dobijamo da je a = −2, b = −8, c = −12, d = 2,

e = 1 i f = 0. Opste resenje date jednacine je zbir yh + yp, odnosno

dvoparametarska familija funkcija

y : x 7→ (C +Dx)ex − 2x2 − 8x− 12 + 2x2ex + cos x, C,D ∈ R.

3. Ako je x = 0, onda je y(t) = Ae−t2/2 (A ∈ R), a ako je y = 0,

onda je x(t) = Be−t2/2, B ∈ R. Za xy 6= 0 iz simetricnog oblika sistema

dx

x2y + tx=

dy

xy2 − ty=dt

1,

dobijamo da jed(xy)

2(xy)2=dt

1,

dx

x(xy + t)=dt

1,

odakle sledi da je

1

xy+ 2t = C,

x

ye−t

2

= D, C,D ∈ R.

72 Prvi kolokvijum

Ako su ϕ i ψ funkcije definisane jednakostima

ϕ(x, y) =1

xy+ 2t, ψ(x, y) =

x

ye−t

2

,

onda je J [ϕ, ψ] =2e−t

2

xy36= 0 za xy 6= 0, pa je sistemom

ϕ(x, y) = C, ψ(x, y) = D, C,D ∈ R

definisano resenje datog sistema.

Primer 2


dx

dt=x(y − t)t(x− y)

dy

dt=y(t− x)t(x− y)


y′′ + 4y′ + 4y = e−2x ln x.


(2xy + 3)dy − y2dx = 0

koje zadovoljava uslov: y(2) = 1/2.

Resenja

1. Ako dati sistem napisemo u simetricnom obliku

dx

x(y − t)=

dy

y(t− x)=

dt

t(x− y),

lako se uocava da je

dx + dy + dt = 0

Prvi kolokvijum 73

i

ytdx + xtdy + xydt = 0,

odnosno d(xyt) = 0. Prvi integrali sistema su

x + y + t = C1, xyz = C2.

Obzirom da je

D(ϕ1, ϕ2)

D(x, y)=

∣

∣

∣

∣

1 1

yt xt

∣

∣

∣

∣

= t(x− y) 6= 0

u oblasti u kojoj je sistem zadat, prvi integrali su nezavisni i odre -duju

resenje sistema.

2. Jednacina moze da se resi metodom varijacije konstanti. Linearno

nezavisna resenja odgovarajuce homogene jednacine su y1(x) = e−2x i

y2(x) = xe−2x. Za odre -divanje funkcija C′1(x) i C′2(x) dobija se sistem

C′1(x) + xC′2(x) = 0

C′1(x) + (1− 2x)C′2(x) = ln x

cije je resenje C′1(x) = −x ln x, C′2(x) = ln x. Integracijom dobijamo da

je

C1(x) = −∫

x ln xdx = −x2

2ln x +

x2

4+D1,

C2(x) =

∫

ln xdx = x ln x− x +D2.

Prema tome, opste resenje date jednacine je

y = (D1 +D2x)e−2x +

x2

2e−2x

(

ln x− 3

2

)

.

3. Ocigledno je y = 0 integralna kriva. Za y 6= 0 data jednacina se

moze napisati u oblikudx

dy=

2xy + 3

y2,

odnosno u obliku

x′y −2

yx =

3

y2.

Dobijena jednacina je linearna po x. Njenim resavanjem dobijamo opste

resenje date jednacine:

x = Cy2 − 1

y.

74 Prvi kolokvijum

Primer 3

1. Odrediti opste resenje sistema diferencijalnih jednacina

x′ − y′ = 2x+ t− 1

y′ + x = y.


y′ =(x+ y + 1)ex − ey

(x+ y + 1)ey − ex.


dx

xy=

dy√

y2 − 1=

dz

(x2 − z)y.

Primer 4

1. Odrediti opste resenje sistema diferencijalnih jednacina

x′ + y′ = − 2y

y′ + 2x′ + 5x = t+ 2.

2. Odrediti partikularno resenje jednacine

yy′′ = (y′)2 − (y′)3

koje zadovoljava uslove y(1) = 1 i y′(1) = −1.


dx

x+ y − xy2=

dy

x2y − x− y=

dz

y2 − x2.

Prvi kolokvijum 75

Primer 5

1. Odrediti partikularno resenje diferencijalne jednacine

(2y − x+ 1)dx+ (2x− 4y + 1)dy = 0


2. Odrediti opste resenje diferencijalne jednacine

y′′ + y′(y′ + 2e−y/2√

y′) = 0.


x′ = y + t, y′ = −x.

Primer 6


(4x− 2y + 1)dx+ (y − 2x− 3)dy = 0



y′′ − y′ = 2ex/2√

y′.


x′ = 3x− 2y + et, y′ = 2x− y.

Primer 7


y′ =y + 2

x+ 1+ tan

y − 2x

x+ 1.

76 Prvi kolokvijum


y′′ + 6y′ + 9y = 6xe3x + 18

koje zadovoljava uslove y(0) = 2 i y′(0) = 0.


dx

xz=

dy

yz − x=dz

z2.

Primer 8


2y − x3

x3y′ =

3y2 + 2x

x4.


y′′ − 4y′ + 5y = e2x + 5x2 + x− 2

koje zadovoljava uslove y(0) = 0 i y′(0) = 3.


dx

x√y=

dy

−2(xy + 2y√y)

=dz

x.

Primer 9

1. Resiti diferencijalnu jednacinu

y′ +x

1− x2y = x

√y.


u′′′ + u′′2

= 0.

Prvi kolokvijum 77

3. Odrediti partikularno resenje sistema

x′ + y′ = 3x+ 16tet

x′ − y′ = − x+ 4y + 16tet

koje zadovoljava uslove x(0) = 0 i y(0) = 1.

Primer 10


z′ +x

1 + x2z = −x

√z.


u′′′ − u′′2

= 0.

3. Odrediti partikularno resenje sistema

y′ − x′ = 3x− 5y + t

x′ + y′ = x+ 3y

koje zadovoljava uslove x(0) = 1 i y(0) = 0.

Primer 11


y′ = − x2 + y2 + y

2xy + x+ ey



y′′′ − 2y′′ + 5y′ = 2et + 3t− 1.

78 Prvi kolokvijum

3. Resiti sistemdx

x=dy

y=

dz

z + u=du

xy.

Primer 12


(2x2 − y2)dx+ 2xydy = 0



y′′′ − y′′ + y′ + y = sin x+ 3xex.

3. Resiti sistem

dx

x2z=

dy

y2z=

dz

x+ y=

du

x+ y.

Primer 13


y′′′ − 3y′′ + 2y′ = 2xex.

2. Dat je sistem diferencijalnih jednacina

u′ = −v + 8t, v′ = 5u− v.

(1) Odrediti fundamentalnu matricu sistema.

(2) Odrediti opste resenje sistema.

3. Odrediti resenje sistema diferencijalnih jednacina

dx

x− y=

dy

x− z=

dz

z − y

Prvi kolokvijum 79

koje zadovoljava uslov x(1) =√2, z(1) = 1.

Primer 14


y′′′ − 2y′′ + y′ = xe2x.

2. Dat je sistem diferencijalnih jednacina

u′ = 2u− 4v + 2, v′ = 5u− 2v − 8t.

(1) Odrediti fundamentalnu matricu sistema.

(2) Odrediti opste resenje sistema.

3. Odrediti resenje sistema diferencijalnih jednacina

dx

xz=

dy

z − y=

dz

y + z

koje zadovoljava uslov x(1) = 1, z(1) =√2.

Primer 15


y′ =y

x2 ln y − x

koje zadovoljava uslov: y(1/2) = 1.

2. (1) Dokazati da su resenja y1(x) i y2(x) jednacine

y′′ + p(x)y′ + q(x)y = 0,

gde su p(x) i q(x) neprekidne funkcije, linearno nezavisnana (a, b) ako i samo ako jeW (y1(x), y2(x)) 6= 0 za x ∈ (a, b).

80 Prvi kolokvijum

(2) Ispitati da li je y(x) = C1x+C+2/x3 opste resenje jednacinex2y′′ + 3xyp − 3y = 0 za x ∈ (0,+∞).


dx

y − x=

dy

x+ y + z=

dz

x− y.

Primer 16


dy

dx=

2x− 3z

5z − 2y,

dz

dx=

3y − 5x

5z − 2y.

2. Resiti jednacinu

xdx+ ydy√

x2 + y2+dy

x=ydx

x2.

3. (1) Metodom varijacije konstanti izvesti opste resenje jednaciney′′ + p(x)y′ + q(x)y = f(x).

(2) Resiti jednacinu y′′ + y = sin x+1

sin x.

Primer 17

1. Resiti jednacinu yy′′ = 1 + y′2.

2. (1) Ako su parcijalni izvodi funkcija P : R2 → R i Q : R2 →R neprekidni, dokazati da je izraz P (x, y)dx + Q(x, y)dytotalni diferencijal ako i samo ako je P ′y = Q′x.

(2) Resiti jednacinu y′(ey + x+ sin x) + ex + y + y cos x = 0.


dy

z + u=

dz

u+ y=

du

y + z=dx

x.

Prvi kolokvijum 81

Primer 18


dx

dt=x(y − t)t(x+ t)

,dy

dt=t2 + xy

t(x+ t).

2. Resiti jednacinu y′(y2 − x2 − 2xy) + y2 − x2 + 2xy = 0.

3. (1) Dokazati da je y(x) = C1y1(x) + C2y2(x) + yp(x) opsteresenje jednacine

y′′ + a(x)y′ + b(x)y = c(x),

gde su y1 i y2 dva linearno nezavisna resenja odgovarajucehomogene jednacine, yp partikularno resenje date jednaci-ne, a C1 i C2 proizvoljne realne konstante.

(2) Resiti jednacinu y′′ + y = π cos x.

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