книги / Математика. Дифференциальные уравнения
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3.27.y′− y = x + 1x ex
3.28.xy′+ y = 4x3
3.29.y′(1− x2 ) + xy2 = 2x (1− x2 )
3.30.y′− x y+1 = ex (x +1)
Задача 4. Найти решение задачи Коши.
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4.01. |
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4.02. |
dx +(xy − y3 )dy = 0 |
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4.03. y =(x2 ln y − x) y′ |
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4.04. |
(2xy +3)dy − y2dx = 0 |
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4.05. |
ydx +(4ln y −2x − y)dy = 0 |
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4.06. |
( y4 + 2x) y′= y |
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4.07. |
( y4ey + 2x) y′= y |
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4.08. |
(xcos2 y − y2 ) y′= y cos2 y |
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4.09.(2 y ln y + y − x) y′= y
4.10.y′(2x − y2 ) =1
4.11.y(1+ y2 )dx =(x + xy2 − y2 )dy
4.12.dy(xcos y +sin 2 y) = dx
4.13.2xy( y2 + 4)dy + y2 ( y2 + 4)dx = 2dy
y(1) = e
y(1) = 0 y(0) = 23 y(0) =1
y(e) =1
y(−1) = 0 y(1) =1 y(0) =1 y(0) =1 y(0) = 2 y(0) =1
y(π) = π4 y(2) =1
y(1) = 0
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y(0) = π
y π = 28
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4.14. |
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y +7)dy = 2 y ydx |
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4.15. |
(2 y + x tgy − y tgy)dy = dx |
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4.16. |
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dy = ydx |
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4.17.(xy + y )dy + y2dx = 0
4.18.(cos 2 y cos2 y − x) y′=sin y cos y
4.19.ydy = ey2 (dx −2xydy)
4.20.(4 y cos 2 y − x) y′= y
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4.21.e2 y + x dy + 4 y2dx = 0
4.22. |
y2 +(xy −1) y′= 0 |
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4.23. |
(sin2 y + x ctgy) y′=1 |
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4.24. |
2ln ydy = ydx − xdy |
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4.25. |
(sin2 2 y −2sin2 y + 2x) y′=sin 2 y |
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4.26. |
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4.27.y3 ( y −2)dx +3xy2 ( y −2)dy =( y +3)dy
4.28.( y3 + xy − y)dy = 12 dx
4.29.ydx +(sin 2 y −2cos2 y + x)dy = 0
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y(−4) =1 y(0) = π
y(e) =1
y −1 = 42
y π =14
y(0) = 0
y(16) = π4 y(e) = 12
y(2) =1 y(1) = π2
y (4) = e2
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y(e) =1
y 1 =39
y(−2) = 0 y(1) = π2
4.30. (1+ x shy)dy = chydx
Задача 5. Найти решение задачи Коши.
5.01. |
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5.02. |
y′− xy = −y3e−x2 |
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5.03. |
xy′+ 2 y =3x3 4 |
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5.04. |
xy′+ y = y2 ln x |
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5.05. |
x(x −1) y′+ y3 = xy |
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5.06. |
y′− y tgx = − |
2 y4 sin x |
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5.07. |
x cos2 x y′+ 2 y cos2 x = 2x y |
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5.08. |
y′+ y = e |
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5.09.2 y′+ y cos x = y−1 cos x(1+sin x)
5.10.2( y′+ xy) =( x −1)ex y2
5.11.4y′+ x3 y =(x3 +8)e−2 x y2
5.12.xy′+ 2 y + x5 y3ex = 0
5.13.y′− y tgx + y2 cos x = 0
5.14.2 y′+3y cos x = e2 x (2 +3cos x) y−1
5.15.3( xy′+ y) = xy2
5.16.8xy′−12 y = −(5x2 +3) y3
5.17.y′+ xy =(1+ x)e−x y2
y(1) = ln 2
y(0) =1
y(0) =1 y(1) =1
y(1) = 12 y(1) =1
y(0) =1
y(2π) =1
y(0) = 94 y(0) =1
y(0) = 2 y(0) =1
y(1) = 12 y(π) = −1
y(0) = 2 y(1) =3 y (1) = 2 y(0) =1
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5.18.y′+ 4x3 y = 4y2e4 x (1− x3 )
5.19.xy′− y = −y2 (ln x + 2)ln x
5.20.3y′+ 2xy = 2xy−2e−2 x2
5.21.2xy′−3y +(20x2 +12) y3 = 0
5.22.3xy′+5y =(4x −5) y4
5.23.y′+ 2xy = 2x3 y3
5.24.x2 y2 y′+ xy3 =1
5.25.2 y′−3y cos x = −e−2 x (2 +3cos x) y−1
5.26. хy′−2 x3 y = y
5.27.( x +1) y′+ y + y2 (x +1) = 0
5.28.y′= x y + x2xy−1
5.29.yy′= xe2 x + y2
5.30.(1− x2 ) y′− xy = 2xy2
y(0) =1 y(1) =1 y(0) = −1
y(1) = 2 12
y(1) =1 y (0) = 2 y(1) =1 y(0) =1 y(2) =8 y(0) = 2
y( 2 ) =1
y(0) = 2 y(0) =1
Задача 6. Найти общий интеграл дифференциального уравнения.
6.1. (x +ex + y)dx +(x + y2 −4)dy = 0
6.2. y′= |
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6.3. |
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6.4. y′= |
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6.5. (2x − y +1)dx +(2 y − x −1)dy = 0
′= x − yex
6.6. y ex − y
6.7. (2x + y +3x2 sin y)dx +(x + x3 cos y + 2 y)dy = 0
6.8. y′= x +ex + 2 y 4 −2x − y2
6.9. (x + 2 −3y2 )dy =(6x2 − y +3)dx
6.10. y′= x2 −3y2 6xy +1
6.11. ey dx +(3y + xey )dy = 0
6.12. y′= y2 + 2 y −2xy
6.13.(cos y − x3 )dy =(3x2 y +sin x)dx
6.14.y′= sin x −2xy
x2 −cos y
6.15. xy dx +( y2 +ln x)dy = 0
6.16. y′= y2 − y + 2 x −2xy
6.17. y cos x dx +sin x dy = 0
6.18. y′= 2 y e−yxey
6.19. (ex + y +sin y)dx +(ey + x + xcos y)dy = 0
6.20. y′= x3 + y y − x
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6.21.(ln y −2x)dx + xy −2 y dy = 0
6.22. |
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8y −10xy +1 |
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6.23. |
(x2 + y2 + y)dx +(2xy + x +ey )dy = 0 |
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6.24. |
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6.25. (x2 + xy2 )dx +(x2 y + y3 )dy = 0 |
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6.26. y′= |
3x2 −6xy2 |
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6x2 y + 4 y3 |
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6.27. ( y3 −ln x)dy = xy dx
6.28. y′= |
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6.29. (x2 + y −5)dx +(x + y +ey )dy = 0
′= cos x − y3 6.30. y 3xy2 +ey
Задача 7. Найти общий интеграл дифференциального уравнения.
7.1.(1− x2 y)dx + x2 ( y − x)dy = 0
7.2.3ey dx +(xey −1)dy = 0
7.3.xdy =(x2 y + y)dx
7.4.(x3 + xy + 2x2 y3 )dx +(x2 +3x3 y2 )dy = 0
7.5.( x − xy)dx +( y + x2 )dy = 0
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7.6.y2dx +(xy −2)dy = 0
7.7.(1+3x2 sin y)dx = xctgy dy
7.8.(3x2 + y2 )dx + xydy = 0
7.9.( x + 2xy)dx +(x −2)dy = 0
7.10.(4xy −3)dx +(x2 +1)dy = 0
7.11.ydx +(2x + 2 y2 sin 2 y −3y cos 2 y)dy = 0
7.12.sin 2 ydx + 2(x −cos2 y cos 2 y)dy = 0
7.13.x(1 − y)dx +(x2 + y)dy = 0
7.14.(x2 + y)dx −2xydy = 0
7.15.(1+ y2 )dx + xydy = 0
7.16.sin ydy =(x −cos y)dx
7.17.(xy2 −2 y3 )dx +(3 −2xy2 )dy = 0
7.18.(2 y + xy3 )dx +(x + x2 y2 )dy = 0
7.19.ydx + xln xdy = 0
7.20.(xy2 −3y3 )dx +(1−3xy2 )dy = 0
7.21.xdy =( y + xy2 )dx
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7.22. |
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7.23.(x + x2 y)dy = ydx
7.24.(x2 y3 + y)dx +(x3 y2 − x)dy = 0
7.25.1+ sinx y dx +(xctgy +1)dy = 0
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7.26.xdy = y(1+ xy)dx
7.27.( x + y) ydx +(xy +1)dy = 0
7.28.y2dx = (xy + x3 )dy
7.29.(x2 + 2x + y)dx +(3x2 y − x)dy = 0
7.30.xdy =(x2 + y)dx
Задача 8. Определить виды дифференциальных уравнений
иуказать методы их решения.
8.01.1) (x − y)dx + xdy = 0
2)(1+ x2 ) y′=1− y3
3)(sin2 y + x ctg y) y′=1
4)3x2ey dx +(x3ey −1)dy = 0
5)2( xy′+ y) = y2 ln x
8.02. 1) 1
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2)2( y′+ y) = xy2
3)xydy + xdx = 0
4)y2 + 4 =3xyy′
5)y2dx +(xy −1)dy = 0
8.03.1) y′+ x y+1 =(1+ x)3
2)2x2dy =(xy + y2 )dx
3)y′=3xy + x
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4) xy′− y = −y2 (ln x + 2)ln x |
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5) |
sin y + y sin x + |
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8.04. 1) |
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2)(2ey − x) y′=1
3)x2 y′= xy + y2
4)y′− y = 4xy
5)yy′= xe2 x + y2
8.05.1) xy′=3y − x4 y2
2)(1−2xy) y′= y( y −1)
3)y2 +1 = xyy′
4)( x + y)dx −( y − x)dy = 0
5)xey2 dx +(x2 yey2 + tg2 y)dy = 0
8.06.1) 3( xy′+ y) = xy2
2)xy2 dx − xyx+1 dy = 0
3)dydx − y = y x+1
4)(2 xy − x)dy + ydx = 0
5)2x2 yy′+ y2 = 2
8.07. 1) (x2 −1) y′− x(x2 −1) y = xy
2)(cos(x + y2 ) +sin x)dx + 2 y cos(x + y2 )dy = 0
3)( x − y)dx +( x + y)dy = 0
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4)(x + xy2 )dy + ydx − y2dx = 0
5)( y4ey + 2x) y′= y
8.08.1) dx +(xy − y3 )dy = 0
2)(x2 −1) y′− xy = x3 − x
3)( y2 −2xy)dx + x2dy = 0
4)dy −2xydx = xdx
5)ey dx +(cos y + xey )dy = 0
8.09.1) (5xy2 − x3 )dx +(5x2 y − y)dy = 0
2)2 y′−3y cos x = −e−2 x (2 +3cos x) y−1
3) xdy − ydx = x2 + y2 dx
4)y = 2(1+ x2 y′) + x2 y′
5)(4ey + x) y′= 2
8.10.1) xy2dy =(x3 + y3 )dx
2)(3x + y2 )dy = ydx
3)2 y′+7x =3xy2
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+ex dx − |
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5) x 5 + y2 + y 4 + x2 y′= 0
8.11.1) 2xy′−3y = −(20x2 +12) y3
2)x3dy =(xy2 + y3 )dx
3)dx − y +x 2 dy = ydy
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