Euler's Method, Modified Euler's Method

1. Numerical Methods
Ordinary Differential Equations - 2
Dr. N. B. Vyas
Department of Mathematics,
Atmiya Institute of Tech. and Science,
Rajkot (Gujarat) - INDIA
niravbvyas@gmail.com
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

2. Ordinary Differential Equations
Euler’s Method:
Consider the differential Equation
dy
dx
= f(x, y), y(x0) = y0
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

3. Ordinary Differential Equations
Euler’s Method:
Consider the differential Equation
dy
dx
= f(x, y), y(x0) = y0
The Taylor’s series is
y(x) = y(x0) +
(x − x0)
1!
y (x0) +
(x − x0)2
2!
y (x0) + . . . - - - (1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

4. Ordinary Differential Equations
Euler’s Method:
Consider the differential Equation
dy
dx
= f(x, y), y(x0) = y0
The Taylor’s series is
y(x) = y(x0) +
(x − x0)
1!
y (x0) +
(x − x0)2
2!
y (x0) + . . . - - - (1)
Now substituting h = x1 − x0 in eq (1), we get
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

5. Ordinary Differential Equations
Euler’s Method:
Consider the differential Equation
dy
dx
= f(x, y), y(x0) = y0
The Taylor’s series is
y(x) = y(x0) +
(x − x0)
1!
y (x0) +
(x − x0)2
2!
y (x0) + . . . - - - (1)
Now substituting h = x1 − x0 in eq (1), we get
y(x1) = y(x0) + hy (x0) +
h2
2!
y (x0) + . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

6. Euler’s Method
If h is chosen small enough then we may neglect the second and
higher order term of h.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

7. Euler’s Method
If h is chosen small enough then we may neglect the second and
higher order term of h.
y1 = y0 + hf(x0, y0)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

8. Euler’s Method
If h is chosen small enough then we may neglect the second and
higher order term of h.
y1 = y0 + hf(x0, y0)
Which is Euler’s first approximation.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

9. Euler’s Method
If h is chosen small enough then we may neglect the second and
higher order term of h.
y1 = y0 + hf(x0, y0)
Which is Euler’s first approximation.
The general step for Euler method is
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

10. Euler’s Method
If h is chosen small enough then we may neglect the second and
higher order term of h.
y1 = y0 + hf(x0, y0)
Which is Euler’s first approximation.
The general step for Euler method is
yi+1 = yi + hf(xi, yi) where i = 0, 1, 2....
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

11. Euler’s Method
Ex.: Use Euler’s method to find y(1.6) given that
dy
dx
= xy
1
2 , y(1) = 1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

12. Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 and
dy
dx
= f(x, y) = xy
1
2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

13. Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 and
dy
dx
= f(x, y) = xy
1
2
we take h = 0.2
1st approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

14. Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 and
dy
dx
= f(x, y) = xy
1
2
we take h = 0.2
1st approximation:
y1 = y0 + hf(x0, y0)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

15. Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 and
dy
dx
= f(x, y) = xy
1
2
we take h = 0.2
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.2)(1)(1)
1
2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

16. Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 and
dy
dx
= f(x, y) = xy
1
2
we take h = 0.2
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.2)(1)(1)
1
2
= 1.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

17. Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 and
dy
dx
= f(x, y) = xy
1
2
we take h = 0.2
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.2)(1)(1)
1
2
= 1.2
x1 = x0 + h = 1 + 0.2 = 1.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

18. Euler’s Method
2nd approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

19. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

20. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)
1
2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

21. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)
1
2
= 1.4629
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

22. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)
1
2
= 1.4629
x2 = x1 + h =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

23. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)
1
2
= 1.4629
x2 = x1 + h = 1.2 + 0.2 = 1.4
3rd approximation:
y3 = y2 + hf(x2, y2)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

24. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)
1
2
= 1.4629
x2 = x1 + h = 1.2 + 0.2 = 1.4
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.4629 + (0.2)(1.4)(1.4629)
1
2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

25. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)
1
2
= 1.4629
x2 = x1 + h = 1.2 + 0.2 = 1.4
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.4629 + (0.2)(1.4)(1.4629)
1
2
= 1.8016
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

26. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)
1
2
= 1.4629
x2 = x1 + h = 1.2 + 0.2 = 1.4
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.4629 + (0.2)(1.4)(1.4629)
1
2
= 1.8016
x3 = x2 + h = 1.4 + 0.2 = 1.6
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

27. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)
1
2
= 1.4629
x2 = x1 + h = 1.2 + 0.2 = 1.4
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.4629 + (0.2)(1.4)(1.4629)
1
2
= 1.8016
x3 = x2 + h = 1.4 + 0.2 = 1.6
∴ y(1.6) = 1.8016
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

28. Euler’s Method
Ex.: Using Euler’s method, find y(0.2), given
dy
dx
= y −
2x
y
, y(0) = 1. (Take h = 0.1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

29. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 and
dy
dx
= f(x, y) = y −
2x
y
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

30. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 and
dy
dx
= f(x, y) = y −
2x
y
we take h = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

31. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 and
dy
dx
= f(x, y) = y −
2x
y
we take h = 0.1
1st approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

32. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 and
dy
dx
= f(x, y) = y −
2x
y
we take h = 0.1
1st approximation:
y1 = y0 + hf(x0, y0)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

33. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 and
dy
dx
= f(x, y) = y −
2x
y
we take h = 0.1
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(1 − 0)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

34. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 and
dy
dx
= f(x, y) = y −
2x
y
we take h = 0.1
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(1 − 0)
= 1.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

35. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 and
dy
dx
= f(x, y) = y −
2x
y
we take h = 0.1
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(1 − 0)
= 1.1
x1 = x0 + h = 0 + 0.1 = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

36. Euler’s Method
2nd approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

37. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

38. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

39. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1) 0.1 −
2(0.1)
1.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

40. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1) 0.1 −
2(0.1)
1.1
= 1.1918
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

41. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1) 0.1 −
2(0.1)
1.1
= 1.1918
x2 = x1 + h = 0.1 + 0.1 = 0.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

42. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1) 0.1 −
2(0.1)
1.1
= 1.1918
x2 = x1 + h = 0.1 + 0.1 = 0.2
∴ y(0.2) = 1.1918
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

43. Euler’s Method
Ex.: Use Euler’s method to obtain an approx value
of y(0.4) for the equation
dy
dx
= x + y, y(0) = 1 with
h = 0.1.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

44. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 and
dy
dx
= f(x, y) = x + y
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

45. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 and
dy
dx
= f(x, y) = x + y
1st approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

46. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 and
dy
dx
= f(x, y) = x + y
1st approximation:
y1 = y0 + hf(x0, y0)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

47. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 and
dy
dx
= f(x, y) = x + y
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(0 + 1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

48. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 and
dy
dx
= f(x, y) = x + y
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(0 + 1)
= 1.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

49. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 and
dy
dx
= f(x, y) = x + y
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.1)(0 + 1)
= 1.1
x1 = x0 + h = 0 + 0.1 = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

50. Euler’s Method
2nd approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

51. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

52. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

53. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

54. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

55. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

56. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

57. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

58. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.22 + (0.1)(0.2 + 1.22)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

59. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.22 + (0.1)(0.2 + 1.22)
= 1.362
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

60. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.22 + (0.1)(0.2 + 1.22)
= 1.362
x3 = x2 + h = 0.2 + 0.1 = 0.3
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

61. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.1 + (0.1)f(0.1, 1.1)
= 1.1 + (0.1)(0.1 + 1.1)
= 1.22
x2 = x1 + h = 0.1 + 0.1 = 0.2
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.22 + (0.1)(0.2 + 1.22)
= 1.362
x3 = x2 + h = 0.2 + 0.1 = 0.3
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

62. Euler’s Method
4th approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

63. Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

64. Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.362 + (0.1)f(0.3, 1.362)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

65. Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.362 + (0.1)f(0.3, 1.362)
= 1.362 + (0.1)(0.3 + 1.362)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

66. Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.362 + (0.1)f(0.3, 1.362)
= 1.362 + (0.1)(0.3 + 1.362)
= 1.5282
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

67. Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.362 + (0.1)f(0.3, 1.362)
= 1.362 + (0.1)(0.3 + 1.362)
= 1.5282
x4 = x3 + h = 0.3 + 0.1 = 0.4
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

68. Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.362 + (0.1)f(0.3, 1.362)
= 1.362 + (0.1)(0.3 + 1.362)
= 1.5282
x4 = x3 + h = 0.3 + 0.1 = 0.4
∴ y(0.4) = 1.5282
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

69. Euler’s Method
Ex.: Given
dy
dx
=
y − x
y + x
, y(0) = 1.
Find y(0.1) by Euler’s method in 5 steps.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

70. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 and
dy
dx
= f(x, y) =
y − x
y + x
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

71. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 and
dy
dx
= f(x, y) =
y − x
y + x
1st approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

72. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 and
dy
dx
= f(x, y) =
y − x
y + x
1st approximation:
y1 = y0 + hf(x0, y0)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

73. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 and
dy
dx
= f(x, y) =
y − x
y + x
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.02)
1 − 0
1 + 0
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

74. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 and
dy
dx
= f(x, y) =
y − x
y + x
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.02)
1 − 0
1 + 0
= 1.02
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

75. Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 and
dy
dx
= f(x, y) =
y − x
y + x
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.02)
1 − 0
1 + 0
= 1.02
x1 = x0 + h = 0 + 0.02 = 0.02
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

76. Euler’s Method
2nd approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

77. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

78. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.02 + (0.02)
1.02 − 0.02
1.02 + 0.02
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

79. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.02 + (0.02)
1.02 − 0.02
1.02 + 0.02
= 1.0392
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

80. Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.02 + (0.02)
1.02 − 0.02
1.02 + 0.02
= 1.0392
x2 = x1 + h = 0.02 + 0.02 = 0.04
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

81. Euler’s Method
3rd approximation:
y3 = y2 + hf(x2, y2)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

82. Euler’s Method
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.0392 + (0.02)
1.0392 − 0.04
1.0392 + 0.04
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

83. Euler’s Method
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.0392 + (0.02)
1.0392 − 0.04
1.0392 + 0.04
= 1.0577
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

84. Euler’s Method
3rd approximation:
y3 = y2 + hf(x2, y2)
= 1.0392 + (0.02)
1.0392 − 0.04
1.0392 + 0.04
= 1.0577
x3 = x2 + h = 0.04 + 0.02 = 0.06
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

85. Euler’s Method
4th approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

86. Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

87. Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.0577 + (0.02)
1.0577 − 0.06
1.0577 + 0.06
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

88. Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.0577 + (0.02)
1.0577 − 0.06
1.0577 + 0.06
= 1.0755
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

89. Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.0577 + (0.02)
1.0577 − 0.06
1.0577 + 0.06
= 1.0755
x4 = x3 + h = 0.06 + 0.02 = 0.08
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

90. Euler’s Method
4th approximation:
y4 = y3 + hf(x3, y3)
= 1.0577 + (0.02)
1.0577 − 0.06
1.0577 + 0.06
= 1.0755
x4 = x3 + h = 0.06 + 0.02 = 0.08
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

91. Euler’s Method
5th approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

92. Euler’s Method
5th approximation:
y4 = y3 + hf(x3, y3)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

93. Euler’s Method
5th approximation:
y4 = y3 + hf(x3, y3)
= 1.0755 + (0.02)
1.0755 − 0.08
1.0755 + 0.08
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

94. Euler’s Method
5th approximation:
y4 = y3 + hf(x3, y3)
= 1.0755 + (0.02)
1.0755 − 0.08
1.0755 + 0.08
= 1.0928
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

95. Euler’s Method
5th approximation:
y4 = y3 + hf(x3, y3)
= 1.0755 + (0.02)
1.0755 − 0.08
1.0755 + 0.08
= 1.0928
x5 = x4 + h = 0.08 + 0.02 = 1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

96. Euler’s Method
5th approximation:
y4 = y3 + hf(x3, y3)
= 1.0755 + (0.02)
1.0755 − 0.08
1.0755 + 0.08
= 1.0928
x5 = x4 + h = 0.08 + 0.02 = 1
∴ y(1) = 1.0928
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

97. Euler’s Method
Ex.: Find y(2) for
dy
dx
=
y
x
, y(1) = 1.
using Euler’s method, take h = 0.2.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

98. Ordinary Differential Equations
Modified Euler’s Method:
By Euler’s method
y1 = y0 + hf(x0, y0)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

99. Ordinary Differential Equations
Modified Euler’s Method:
By Euler’s method
y1 = y0 + hf(x0, y0)
For better approximation y
(1)
1 of y1, we take
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

100. Ordinary Differential Equations
Modified Euler’s Method:
By Euler’s method
y1 = y0 + hf(x0, y0)
For better approximation y
(1)
1 of y1, we take
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

101. Ordinary Differential Equations
Modified Euler’s Method:
By Euler’s method
y1 = y0 + hf(x0, y0)
For better approximation y
(1)
1 of y1, we take
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
where x1 = x0 + h
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

102. Ordinary Differential Equations
Modified Euler’s Method:
By Euler’s method
y1 = y0 + hf(x0, y0)
For better approximation y
(1)
1 of y1, we take
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
where x1 = x0 + h
For still better approximation y
(2)
1 of y1,
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

103. Ordinary Differential Equations
Modified Euler’s Method:
By Euler’s method
y1 = y0 + hf(x0, y0)
For better approximation y
(1)
1 of y1, we take
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
where x1 = x0 + h
For still better approximation y
(2)
1 of y1,
y
(2)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(2)
1 )
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

104. Ordinary Differential Equations
Modified Euler’s Method:
By Euler’s method
y1 = y0 + hf(x0, y0)
For better approximation y
(1)
1 of y1, we take
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
where x1 = x0 + h
For still better approximation y
(2)
1 of y1,
y
(2)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(2)
1 )
we repeat this process till two consecutive values
of y agree.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

105. Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,
we find y2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

106. Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,
we find y2
y2 = y1 + hf(x1, y1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

107. Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,
we find y2
y2 = y1 + hf(x1, y1)
For better approximation y
(1)
2 of y2, we take
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

108. Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,
we find y2
y2 = y1 + hf(x1, y1)
For better approximation y
(1)
2 of y2, we take
y
(1)
2 = y1 +
h
2
[f(x1, y1) + f(x2, y2)]
where x2 = x1 + h
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

109. Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,
we find y2
y2 = y1 + hf(x1, y1)
For better approximation y
(1)
2 of y2, we take
y
(1)
2 = y1 +
h
2
[f(x1, y1) + f(x2, y2)]
where x2 = x1 + h
For still better approximation y
(2)
2 of y2,
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

110. Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,
we find y2
y2 = y1 + hf(x1, y1)
For better approximation y
(1)
2 of y2, we take
y
(1)
2 = y1 +
h
2
[f(x1, y1) + f(x2, y2)]
where x2 = x1 + h
For still better approximation y
(2)
2 of y2,
y
(2)
2 = y1 +
h
2
f(x1, y1) + f(x2, y
(2)
2 )
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

111. Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,
we find y2
y2 = y1 + hf(x1, y1)
For better approximation y
(1)
2 of y2, we take
y
(1)
2 = y1 +
h
2
[f(x1, y1) + f(x2, y2)]
where x2 = x1 + h
For still better approximation y
(2)
2 of y2,
y
(2)
2 = y1 +
h
2
f(x1, y1) + f(x2, y
(2)
2 )
we repeat this step until y2 becomes stationary.
Then we proceed to calculate y3 in the same way
as above.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

112. Modified Euler’s Method
Ex.: Solve
dy
dx
= x + y , y(0) = 1.
by Euler’s modified method for x = 0.1
correct upto four decimal places by taking h = 0.05.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

113. Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.05 and
dy
dx
= f(x, y) = x + y
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

114. Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.05 and
dy
dx
= f(x, y) = x + y
x1 = x0 + h = 0 + 0.05 = 0.05
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

115. Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.05 and
dy
dx
= f(x, y) = x + y
x1 = x0 + h = 0 + 0.05 = 0.05
y1 = y0 + hf(x0, y0)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

116. Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.05 and
dy
dx
= f(x, y) = x + y
x1 = x0 + h = 0 + 0.05 = 0.05
y1 = y0 + hf(x0, y0)
= 1 + (0.05)(1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

117. Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.05 and
dy
dx
= f(x, y) = x + y
x1 = x0 + h = 0 + 0.05 = 0.05
y1 = y0 + hf(x0, y0)
= 1 + (0.05)(1)
= 1.05
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

118. Modified Euler’s Method
1st approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

119. Modified Euler’s Method
1st approximation:
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

120. Modified Euler’s Method
1st approximation:
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
= 1 + 0.05
2 [(0 + 1) + (0.05 + 1.05)] =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

121. Modified Euler’s Method
1st approximation:
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
= 1 + 0.05
2 [(0 + 1) + (0.05 + 1.05)] = 1.0525
2nd approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

122. Modified Euler’s Method
1st approximation:
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
= 1 + 0.05
2 [(0 + 1) + (0.05 + 1.05)] = 1.0525
2nd approximation:
y
(2)
1 = y0 +
h
2
f(x0, y0) +
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

123. Modified Euler’s Method
1st approximation:
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
= 1 + 0.05
2 [(0 + 1) + (0.05 + 1.05)] = 1.0525
2nd approximation:
y
(2)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(1)
1 )
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

124. Modified Euler’s Method
1st approximation:
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
= 1 + 0.05
2 [(0 + 1) + (0.05 + 1.05)] = 1.0525
2nd approximation:
y
(2)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(1)
1 )
= 1 + 0.05
2 [(0 + 1) + (0.05 + 1.0525)] =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

125. Modified Euler’s Method
1st approximation:
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
= 1 + 0.05
2 [(0 + 1) + (0.05 + 1.05)] = 1.0525
2nd approximation:
y
(2)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(1)
1 )
= 1 + 0.05
2 [(0 + 1) + (0.05 + 1.0525)] = 1.05256
∴ y1 = 1.05256 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

126. Modified Euler’s Method
x2 = x1 + h = 0.05 + 0.05 = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

127. Modified Euler’s Method
x2 = x1 + h = 0.05 + 0.05 = 0.1
y2 = y1 + hf(x1, y1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

128. Modified Euler’s Method
x2 = x1 + h = 0.05 + 0.05 = 0.1
y2 = y1 + hf(x1, y1)
= 1.05256 + (0.05)(0.1 + 1.05256)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

129. Modified Euler’s Method
x2 = x1 + h = 0.05 + 0.05 = 0.1
y2 = y1 + hf(x1, y1)
= 1.05256 + (0.05)(0.1 + 1.05256)
= 1.10769
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

130. Modified Euler’s Method
1st approximation:
y
(1)
2 = y1 +
h
2
[f(x1, y1) + f(x2, y2)]
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

131. Modified Euler’s Method
1st approximation:
y
(1)
2 = y1 +
h
2
[f(x1, y1) + f(x2, y2)]
=
1.05256 + 0.05
2 [(0.05 + 1.05256) + (0.1 + 1.10769)]
=
2nd approximation:
y
(2)
2 = y1 +
h
2
f(x1, y1) + f(x2, y
(1)
2 )
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

132. Modified Euler’s Method
1st approximation:
y
(1)
2 = y1 +
h
2
[f(x1, y1) + f(x2, y2)]
=
1.05256 + 0.05
2 [(0.05 + 1.05256) + (0.1 + 1.10769)]
=
2nd approximation:
y
(2)
2 = y1 +
h
2
f(x1, y1) + f(x2, y
(1)
2 )
∴ y2 = .... correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

133. Modified Euler’s Method
Ex.: Using modified Euler’s method , find y(0.2)
and y(0.4) given that
dy
dx
= y + ex
, y(0) = 0
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

134. Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 0, h = 0.2 and
dy
dx
= f(x, y) = y + ex
x1 = x0 + h = 0.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

135. Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 0, h = 0.2 and
dy
dx
= f(x, y) = y + ex
x1 = x0 + h = 0.2
y1 = y0 + hf(x0, y0)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

136. Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 0, h = 0.2 and
dy
dx
= f(x, y) = y + ex
x1 = x0 + h = 0.2
y1 = y0 + hf(x0, y0)
= 0 + (0.2)(0 + e0
)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

137. Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 0, h = 0.2 and
dy
dx
= f(x, y) = y + ex
x1 = x0 + h = 0.2
y1 = y0 + hf(x0, y0)
= 0 + (0.2)(0 + e0
)
= 0.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

138. Modified Euler’s Method
1st approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

139. Modified Euler’s Method
1st approximation:
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

140. Modified Euler’s Method
1st approximation:
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
= 0 + 0.2
2 [f(0, 0) + f(0.2, 0.2)] =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

141. Modified Euler’s Method
1st approximation:
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
= 0 + 0.2
2 [f(0, 0) + f(0.2, 0.2)] = 0.24214
2nd approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

142. Modified Euler’s Method
1st approximation:
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
= 0 + 0.2
2 [f(0, 0) + f(0.2, 0.2)] = 0.24214
2nd approximation:
y
(2)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(1)
1 )
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

143. Modified Euler’s Method
1st approximation:
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
= 0 + 0.2
2 [f(0, 0) + f(0.2, 0.2)] = 0.24214
2nd approximation:
y
(2)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(1)
1 )
= 0 + 0.2
2 [f(0, 0) + f(0.2, 0.24214)] =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

144. Modified Euler’s Method
1st approximation:
y
(1)
1 = y0 +
h
2
[f(x0, y0) + f(x1, y1)]
= 0 + 0.2
2 [f(0, 0) + f(0.2, 0.2)] = 0.24214
2nd approximation:
y
(2)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(1)
1 )
= 0 + 0.2
2 [f(0, 0) + f(0.2, 0.24214)] = 0.24635
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

145. Modified Euler’s Method
3rd approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

146. Modified Euler’s Method
3rd approximation:
y
(3)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(2)
1 )
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

147. Modified Euler’s Method
3rd approximation:
y
(3)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(2)
1 )
= 0 + 0.2
2 [f(0, 0) + f(0.2, 0.24635)] =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

148. Modified Euler’s Method
3rd approximation:
y
(3)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(2)
1 )
= 0 + 0.2
2 [f(0, 0) + f(0.2, 0.24635)] = 0.24678
4th approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

149. Modified Euler’s Method
3rd approximation:
y
(3)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(2)
1 )
= 0 + 0.2
2 [f(0, 0) + f(0.2, 0.24635)] = 0.24678
4th approximation:
y
(4)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(3)
1 )
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

150. Modified Euler’s Method
3rd approximation:
y
(3)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(2)
1 )
= 0 + 0.2
2 [f(0, 0) + f(0.2, 0.24635)] = 0.24678
4th approximation:
y
(4)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(3)
1 )
= 0 + 0.2
2 [f(0, 0) + f(0.2, 0.24678)] =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

151. Modified Euler’s Method
3rd approximation:
y
(3)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(2)
1 )
= 0 + 0.2
2 [f(0, 0) + f(0.2, 0.24635)] = 0.24678
4th approximation:
y
(4)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(3)
1 )
= 0 + 0.2
2 [f(0, 0) + f(0.2, 0.24678)] = 0.24681
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

152. Modified Euler’s Method
5th approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

153. Modified Euler’s Method
5th approximation:
y
(5)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(4)
1 )
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

154. Modified Euler’s Method
5th approximation:
y
(5)
1 = y0 +
h
2
f(x0, y0) + f(x1, y
(4)
1 )
= 0.24682
∴ y1 = 0.24682 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

155. Modified Euler’s Method
x2 = x1 + h = 0.4
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

156. Modified Euler’s Method
x2 = x1 + h = 0.4
y2 = y1 + hf(x1, y1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

157. Modified Euler’s Method
x2 = x1 + h = 0.4
y2 = y1 + hf(x1, y1)
= 0.24682 + (0.2)f(0.2, 0.24682)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

158. Modified Euler’s Method
x2 = x1 + h = 0.4
y2 = y1 + hf(x1, y1)
= 0.24682 + (0.2)f(0.2, 0.24682)
= 0.54046
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

159. Modified Euler’s Method
1st approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

160. Modified Euler’s Method
1st approximation:
y
(1)
2 = y1 +
h
2
[f(x1, y1) + f(x2, y2)]
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

161. Modified Euler’s Method
1st approximation:
y
(1)
2 = y1 +
h
2
[f(x1, y1) + f(x2, y2)]
= 0.59687
2nd approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

162. Modified Euler’s Method
1st approximation:
y
(1)
2 = y1 +
h
2
[f(x1, y1) + f(x2, y2)]
= 0.59687
2nd approximation:
y
(2)
2 = y1 +
h
2
f(x1, y1) + f(x2, y
(1)
2 )
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

163. Modified Euler’s Method
1st approximation:
y
(1)
2 = y1 +
h
2
[f(x1, y1) + f(x2, y2)]
= 0.59687
2nd approximation:
y
(2)
2 = y1 +
h
2
f(x1, y1) + f(x2, y
(1)
2 )
=0.60251
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

164. Modified Euler’s Method
3rd approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

165. Modified Euler’s Method
3rd approximation:
y
(3)
2 = y1 +
h
2
f(x1, y1) + f(x2, y
(2)
2 )
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

166. Modified Euler’s Method
3rd approximation:
y
(3)
2 = y1 +
h
2
f(x1, y1) + f(x2, y
(2)
2 )
= 0.60308
4th approximation:
y
(4)
2 = y1 +
h
2
f(x1, y1) + f(x2, y
(3)
2 )
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

167. Modified Euler’s Method
3rd approximation:
y
(3)
2 = y1 +
h
2
f(x1, y1) + f(x2, y
(2)
2 )
= 0.60308
4th approximation:
y
(4)
2 = y1 +
h
2
f(x1, y1) + f(x2, y
(3)
2 )
= 0.60313
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

168. Modified Euler’s Method
5th approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

169. Modified Euler’s Method
5th approximation:
y
(5)
2 = y1 +
h
2
f(x1, y1) + f(x2, y
(4)
2 )
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

170. Modified Euler’s Method
5th approximation:
y
(5)
2 = y1 +
h
2
f(x1, y1) + f(x2, y
(4)
2 )
= 0.60314
∴ y2 = 0.60314 correct up to 4 decimal places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -