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 = sin 3x, 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 = –