Задание 2

Найти частное решение дифференциального уравнения:

1. xdy – ylnydx = 0 , x= 1,y =e (у=е^2х)

2. ху=х+у,у(1) = 0 , (у=xln|x| )

3. , y(1) = 3

4. ,у(0) = 0 ()

5. 

6. y – y = ,x = 0, y = 1

7. (1+x²)y = 2xy, y(2) = 1 (y = 0,2(x² + 1) )

8. xy – y = ylny – ylnx, у(1) = е² ( y = xe^2x )

9. 3y y² – y³ = x, у(0) = 2

10. x(y²+ 1) +y(x²– 1)y= 0 ,у(2) = 1

11. ,х=е,у= 0,

12. , x = 2, y = 4

13. tgx dy – (y – 1)dx = 0, x =,y = 0 ( y = 1–2sinx )

14. 

15. xdy + (y – xsinx)dx = 0 , y() = 0

16. ()

17. 16. (х–у)dx + xdy= 0,у(1) = 2. (у=2х–хln|x|)

18. y + y – x – 2 = 0 , y(0) =0 (y = e^–x + x – 1)

19. ydy + x e ^y dx = 0, x = 1, y = 0 ( 0,5x² –(y + 1)e^–y = –0,5 )

20. ,

21. ysinx + y cosx = 1 , (y = sinx + (– 1)cosx )

22. ,

23. ,

24. y – y = xe^x , y(0) =1 ( y = e^x(0,5x² + 1) )

25. , у(0) =0

26. ,

27. y (2x + 1) = 4x² + 2y – 1, x = 0, y = 1 ( y =(2x+1)(x+C–ln|2x+1|) )

28. 3xdy – ylnydx = 0 , x = 1, y = e

29. , у(1) = 0

30. y +2xy =x,y(0) = 2

Задание 3

Найти все решения дифференциального уравнения

1. (у(у+1)^–1=С(х–1) ,у= –1)

2. (x + xy)dx + y(1 +x)dy = 0, (x + y =ln(x+1)(y+1)+С,у= –1,х = –1 )

3. (xy²+y²) = (x²y–x²)y=0

4. (xy²+y²) – (x²y–x²)y= 0

5.

6. , (у=x sinln|Cx|,у=х)

7. (y= 0,25xln²|Cx| ,у= 0)

8. (y²– 2xy)dx + x²dy = 0 (ln|Cyx^-2| –x^–1= 0,у=х)

9. (x–y)y –x²y = 0 (y=xln^–1|Cx|,у= 0 )

10.

11. (x – y)ydx = x² dy

12. yy + x = 2y

13. (x² – 4)y – 4y = – (x+2)y²

14. xy +y = xy²lnx

15.

16.

17. y cosx + ysinx = y² (y = –cosx(x+C)^–1, у = 0)

18.

19. y – ytgx = –y²cosx,

20. y – +y² = 0

Задание 4.

Решить дифференциальные уравнения высших порядков:

1. (1 + е^х)у + у = 0 (у= С₁(х–е^–х) + С₂

2. у– 2у(у )³= 0

3. у = 2(у – 1)ctgx

4. у = 5x⁴ – sin3x, y(0) = y(0)=0, y(0)=1

5. 

6. 2yy=

7. y + (y )²= 0 (y=ln|C₁x+C₂|)

8. y(x – 1)² = 3, x = 2, y = y = 0 ( y = –3ln|x–1| +2x – 6)

9. xy = y + x²

10. yy = ( y )²

11. xy +y = 1 + x ( y = x +0,25x² + C₁ln|x| + C₂ )

12. ycos²x = 5 , x = 0, y = 1, y = 0 ( y = –5ln|cosx| +1 )

13. (1–x²)y = 4 + xy ( y = 2arcsin²x + C₁arcsinx +C₂ )

14. y + = 0 (y = sin(x + C₂) –C₁ )

15. (y )²=y x

16. (x² + 1)y = 1 , x = 0, y = 1, y = 0 (y = xarctgx – 0,5ln(1+x²)+C₁x +C₂ )

17. x²y + xy = 1 ( y= xln|x| – x + C₁x + C₂ )

18. y (y – 1) = 2(y )²

19. y = 2(y – 1)ctgx

20. y = sin²x +1, x = ,y = 0, y = .

21. xy + y = x²+ 1

22. yy = 3(y )²

23. 

24. y=sin3x,x= ,y=0 ,y= 0,y= 1

25. y + y tgx = sin2x (y = C₁sinx –x –0,5sin2x + C₂ )

26. yy – (y )² = 1

27. y^IVtgx = y+ 1

28. y = 1 , x =0, y =0, y =1 (y = xarcsinx ++C₁x + C₂ )

29. (1+x²)y + (y )² + 1 = 0 (,указание: arctgx+arctgy=arctg)

30. xy + y + x = 0 ( y = 0,5C₁²ln|x –0,25x² +C₂ )

31. y =

32. y =sin3x +,x = , y = 1, y = 0

33. (1 – x²)y – xy = 2 (y = arcsin²x +Carcsinx + C₁)

34. 2yy = 1 + (y )²

35. y–2yctgx = sin³x

36. y = 24(x + 1)³, x = 0, y = 0, y = 1, y= –1 ( y=0,2(x+1)⁶–3,5x²–0,2x–0,2 )

37. y = –

38. y tg y = 2(y )² ( C₁²x +C₂ +ctgy = 0 )

39. (y=x+C₁xe^x + C₂)

40. y x(x+1)² = 1,x=1,y=0,y = –