The document discusses solving ordinary differential equations using Taylor's series method. It presents the Taylor's series for the first order differential equation dy/dx = f(x,y) and gives an example of solving the equation y = x + y, y(0) = 1 using this method. The solution is obtained by taking the Taylor's series expansion and determining the derivatives of y evaluated at x0 = 0. The values of y are computed at x = 0.1 and x = 0.2. A second example solves the differential equation dy/dx = 3x + y^2 using the same approach.

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

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

3. Ordinary Differential Equations
Taylor’s Series Method:
Consider the first order 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) + . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

4. Taylor’s Series Method
Ex:1
Solve y = x + y, y(0) = 1 by Taylor’s series
method. Hence find values of y at x = 0.1 and
x = 0.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

5. Taylor’s Series Method
Sol.: Here y = f(x, y) = x + y, x0 = 0 and y0 = 1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

6. Taylor’s Series Method
Sol.: Here y = f(x, y) = x + y, x0 = 0 and y0 = 1
y = x + y
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

7. Taylor’s Series Method
Sol.: Here y = f(x, y) = x + y, x0 = 0 and y0 = 1
y = x + y ⇒ y (0) = 1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

8. Taylor’s Series Method
Sol.: Here y = f(x, y) = x + y, x0 = 0 and y0 = 1
y = x + y ⇒ y (0) = 1
y = 1 + y
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

9. Taylor’s Series Method
Sol.: Here y = f(x, y) = x + y, x0 = 0 and y0 = 1
y = x + y ⇒ y (0) = 1
y = 1 + y ⇒ y (0) = 1 + y (0) = 1 + 1 = 2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

10. Taylor’s Series Method
Sol.: Here y = f(x, y) = x + y, x0 = 0 and y0 = 1
y = x + y ⇒ y (0) = 1
y = 1 + y ⇒ y (0) = 1 + y (0) = 1 + 1 = 2
y = y
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

11. Taylor’s Series Method
Sol.: Here y = f(x, y) = x + y, x0 = 0 and y0 = 1
y = x + y ⇒ y (0) = 1
y = 1 + y ⇒ y (0) = 1 + y (0) = 1 + 1 = 2
y = y ⇒ y (0) = y (0) = 2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

12. Taylor’s Series Method
Sol.: Here y = f(x, y) = x + y, x0 = 0 and y0 = 1
y = x + y ⇒ y (0) = 1
y = 1 + y ⇒ y (0) = 1 + y (0) = 1 + 1 = 2
y = y ⇒ y (0) = y (0) = 2
yiv
= y
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

13. Taylor’s Series Method
Sol.: Here y = f(x, y) = x + y, x0 = 0 and y0 = 1
y = x + y ⇒ y (0) = 1
y = 1 + y ⇒ y (0) = 1 + y (0) = 1 + 1 = 2
y = y ⇒ y (0) = y (0) = 2
yiv
= y ⇒ yiv
(0) = y (0) = 2 . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

14. Taylor’s Series Method
Sol.: Here y = f(x, y) = x + y, x0 = 0 and y0 = 1
y = x + y ⇒ y (0) = 1
y = 1 + y ⇒ y (0) = 1 + y (0) = 1 + 1 = 2
y = y ⇒ y (0) = y (0) = 2
yiv
= y ⇒ yiv
(0) = y (0) = 2 . . .
Taylor’s series is
y(x) = y(x0)+
(x − x0)
1!
y (x0)+
(x − x0)2
2!
y (x0)+. . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

15. Taylor’s Series Method
y(x) = 1 + xy (0) +
x2
2!
y (0) +
x3
3!
y (0) . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

16. Taylor’s Series Method
y(x) = 1 + xy (0) +
x2
2!
y (0) +
x3
3!
y (0) . . .
= 1 + x(1) +
x2
2
(2) +
x3
6
(2) +
x4
24
(2) + . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

17. Taylor’s Series Method
y(x) = 1 + xy (0) +
x2
2!
y (0) +
x3
3!
y (0) . . .
= 1 + x(1) +
x2
2
(2) +
x3
6
(2) +
x4
24
(2) + . . .
= 1 + x + x2
+
x3
3
+
x4
12
+ . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

18. Taylor’s Series Method
y(x) = 1 + xy (0) +
x2
2!
y (0) +
x3
3!
y (0) . . .
= 1 + x(1) +
x2
2
(2) +
x3
6
(2) +
x4
24
(2) + . . .
= 1 + x + x2
+
x3
3
+
x4
12
+ . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

19. Taylor’s Series Method
y(x) = 1 + xy (0) +
x2
2!
y (0) +
x3
3!
y (0) . . .
= 1 + x(1) +
x2
2
(2) +
x3
6
(2) +
x4
24
(2) + . . .
= 1 + x + x2
+
x3
3
+
x4
12
+ . . .
y(0.1) = 1 + (0.1) + (0.1)2
+
(0.1)3
3
+
(0.1)4
12
+ . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

20. Taylor’s Series Method
y(x) = 1 + xy (0) +
x2
2!
y (0) +
x3
3!
y (0) . . .
= 1 + x(1) +
x2
2
(2) +
x3
6
(2) +
x4
24
(2) + . . .
= 1 + x + x2
+
x3
3
+
x4
12
+ . . .
y(0.1) = 1 + (0.1) + (0.1)2
+
(0.1)3
3
+
(0.1)4
12
+ . . .
= 1.1103
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

21. Taylor’s Series Method
y(x) = 1 + xy (0) +
x2
2!
y (0) +
x3
3!
y (0) . . .
= 1 + x(1) +
x2
2
(2) +
x3
6
(2) +
x4
24
(2) + . . .
= 1 + x + x2
+
x3
3
+
x4
12
+ . . .
y(0.1) = 1 + (0.1) + (0.1)2
+
(0.1)3
3
+
(0.1)4
12
+ . . .
= 1.1103
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

22. Taylor’s Series Method
y(x) = 1 + xy (0) +
x2
2!
y (0) +
x3
3!
y (0) . . .
= 1 + x(1) +
x2
2
(2) +
x3
6
(2) +
x4
24
(2) + . . .
= 1 + x + x2
+
x3
3
+
x4
12
+ . . .
y(0.1) = 1 + (0.1) + (0.1)2
+
(0.1)3
3
+
(0.1)4
12
+ . . .
= 1.1103
y(0.2) = 1 + (0.2) + (0.2)2
+
(0.2)3
3
+
(0.2)4
12
+ . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

23. Taylor’s Series Method
y(x) = 1 + xy (0) +
x2
2!
y (0) +
x3
3!
y (0) . . .
= 1 + x(1) +
x2
2
(2) +
x3
6
(2) +
x4
24
(2) + . . .
= 1 + x + x2
+
x3
3
+
x4
12
+ . . .
y(0.1) = 1 + (0.1) + (0.1)2
+
(0.1)3
3
+
(0.1)4
12
+ . . .
= 1.1103
y(0.2) = 1 + (0.2) + (0.2)2
+
(0.2)3
3
+
(0.2)4
12
+ . . .
= 1.2428
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

24. Taylor’s Series
Ex Using Taylor’s series method, obtain the solution
of
dy
dx
= 3x + y2
, given that y(0) = 1. Find the
value of y for x = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

25. Taylor’s Series
Sol. Here,y = f(x, y) = 3x + y2
, x0 = 0 and y0 = 1.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

26. Taylor’s Series
Sol. Here,y = f(x, y) = 3x + y2
, x0 = 0 and y0 = 1.
y = 3x + y2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

27. Taylor’s Series
Sol. Here,y = f(x, y) = 3x + y2
, x0 = 0 and y0 = 1.
y = 3x + y2
⇒ y (x0) = 3(x0) + y2
0 = 3(0) + 1 = 1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

28. Taylor’s Series
Sol. Here,y = f(x, y) = 3x + y2
, x0 = 0 and y0 = 1.
y = 3x + y2
⇒ y (x0) = 3(x0) + y2
0 = 3(0) + 1 = 1
y = 3 + 2yy
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

29. Taylor’s Series
Sol. Here,y = f(x, y) = 3x + y2
, x0 = 0 and y0 = 1.
y = 3x + y2
⇒ y (x0) = 3(x0) + y2
0 = 3(0) + 1 = 1
y = 3 + 2yy ⇒ y (x0) = 3 + 2(1)(1) = 5
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

30. Taylor’s Series
Sol. Here,y = f(x, y) = 3x + y2
, x0 = 0 and y0 = 1.
y = 3x + y2
⇒ y (x0) = 3(x0) + y2
0 = 3(0) + 1 = 1
y = 3 + 2yy ⇒ y (x0) = 3 + 2(1)(1) = 5
y = 2(y )2
+ 2yy
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

31. Taylor’s Series
Sol. Here,y = f(x, y) = 3x + y2
, x0 = 0 and y0 = 1.
y = 3x + y2
⇒ y (x0) = 3(x0) + y2
0 = 3(0) + 1 = 1
y = 3 + 2yy ⇒ y (x0) = 3 + 2(1)(1) = 5
y = 2(y )2
+ 2yy ⇒ y (x0) = 2(1)2
+ 2(5) = 12
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

32. Taylor’s Series
Sol. Here,y = f(x, y) = 3x + y2
, x0 = 0 and y0 = 1.
y = 3x + y2
⇒ y (x0) = 3(x0) + y2
0 = 3(0) + 1 = 1
y = 3 + 2yy ⇒ y (x0) = 3 + 2(1)(1) = 5
y = 2(y )2
+ 2yy ⇒ y (x0) = 2(1)2
+ 2(5) = 12
By Taylor’s series,
y(x) = y0+(x−x0)y (x0)+
(x − x0)2
2!
y (x0)+
(x − x0)3
3!
y (x0)+. . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

33. Taylor’s Series
Sol. Here,y = f(x, y) = 3x + y2
, x0 = 0 and y0 = 1.
y = 3x + y2
⇒ y (x0) = 3(x0) + y2
0 = 3(0) + 1 = 1
y = 3 + 2yy ⇒ y (x0) = 3 + 2(1)(1) = 5
y = 2(y )2
+ 2yy ⇒ y (x0) = 2(1)2
+ 2(5) = 12
By Taylor’s series,
y(x) = y0+(x−x0)y (x0)+
(x − x0)2
2!
y (x0)+
(x − x0)3
3!
y (x0)+. . .
= 1 + x +
x2
2!
(5) +
x3
3!
(12) + . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

34. Taylor’s Series
Sol. Here,y = f(x, y) = 3x + y2
, x0 = 0 and y0 = 1.
y = 3x + y2
⇒ y (x0) = 3(x0) + y2
0 = 3(0) + 1 = 1
y = 3 + 2yy ⇒ y (x0) = 3 + 2(1)(1) = 5
y = 2(y )2
+ 2yy ⇒ y (x0) = 2(1)2
+ 2(5) = 12
By Taylor’s series,
y(x) = y0+(x−x0)y (x0)+
(x − x0)2
2!
y (x0)+
(x − x0)3
3!
y (x0)+. . .
= 1 + x +
x2
2!
(5) +
x3
3!
(12) + . . .
= 1 + x +
5x2
2!
+ 2x3
+ . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

35. Taylor’s Series
Sol. Here,y = f(x, y) = 3x + y2
, x0 = 0 and y0 = 1.
y = 3x + y2
⇒ y (x0) = 3(x0) + y2
0 = 3(0) + 1 = 1
y = 3 + 2yy ⇒ y (x0) = 3 + 2(1)(1) = 5
y = 2(y )2
+ 2yy ⇒ y (x0) = 2(1)2
+ 2(5) = 12
By Taylor’s series,
y(x) = y0+(x−x0)y (x0)+
(x − x0)2
2!
y (x0)+
(x − x0)3
3!
y (x0)+. . .
= 1 + x +
x2
2!
(5) +
x3
3!
(12) + . . .
= 1 + x +
5x2
2!
+ 2x3
+ . . .
y(0.1) =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

36. Taylor’s Series
Sol. Here,y = f(x, y) = 3x + y2
, x0 = 0 and y0 = 1.
y = 3x + y2
⇒ y (x0) = 3(x0) + y2
0 = 3(0) + 1 = 1
y = 3 + 2yy ⇒ y (x0) = 3 + 2(1)(1) = 5
y = 2(y )2
+ 2yy ⇒ y (x0) = 2(1)2
+ 2(5) = 12
By Taylor’s series,
y(x) = y0+(x−x0)y (x0)+
(x − x0)2
2!
y (x0)+
(x − x0)3
3!
y (x0)+. . .
= 1 + x +
x2
2!
(5) +
x3
3!
(12) + . . .
= 1 + x +
5x2
2!
+ 2x3
+ . . .
y(0.1) =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

37. Taylor’s Series
Ex Using Taylor’s series method, find the solution of
dy
dx
= 2y + 3ex
, y(0) = 0,at x = 0.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

38. Taylor’s Series
Sol. Here,y = f(x, y) = 2y + 3ex
, x0 = 0 and y0 = 0.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

39. Taylor’s Series
Sol. Here,y = f(x, y) = 2y + 3ex
, x0 = 0 and y0 = 0.
y = 2y + 3ex
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

40. Taylor’s Series
Sol. Here,y = f(x, y) = 2y + 3ex
, x0 = 0 and y0 = 0.
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(0) + 3e0
= 3
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

41. Taylor’s Series
Sol. Here,y = f(x, y) = 2y + 3ex
, x0 = 0 and y0 = 0.
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(0) + 3e0
= 3
y = 2y + 3ex
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

42. Taylor’s Series
Sol. Here,y = f(x, y) = 2y + 3ex
, x0 = 0 and y0 = 0.
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(0) + 3e0
= 3
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(3) + 3e0
= 9
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

43. Taylor’s Series
Sol. Here,y = f(x, y) = 2y + 3ex
, x0 = 0 and y0 = 0.
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(0) + 3e0
= 3
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(3) + 3e0
= 9
y = 2y + 3ex
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

44. Taylor’s Series
Sol. Here,y = f(x, y) = 2y + 3ex
, x0 = 0 and y0 = 0.
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(0) + 3e0
= 3
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(3) + 3e0
= 9
y = 2y + 3ex
⇒ y (x0) = 2(y0 ) + 3ex0
= 2(9) + 3e0
= 21
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

45. Taylor’s Series
Sol. Here,y = f(x, y) = 2y + 3ex
, x0 = 0 and y0 = 0.
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(0) + 3e0
= 3
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(3) + 3e0
= 9
y = 2y + 3ex
⇒ y (x0) = 2(y0 ) + 3ex0
= 2(9) + 3e0
= 21
yiv
= 2y + 3ex
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

46. Taylor’s Series
Sol. Here,y = f(x, y) = 2y + 3ex
, x0 = 0 and y0 = 0.
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(0) + 3e0
= 3
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(3) + 3e0
= 9
y = 2y + 3ex
⇒ y (x0) = 2(y0 ) + 3ex0
= 2(9) + 3e0
= 21
yiv
= 2y + 3ex
⇒ yiv
(x0) = 45
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

47. Taylor’s Series
Sol. Here,y = f(x, y) = 2y + 3ex
, x0 = 0 and y0 = 0.
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(0) + 3e0
= 3
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(3) + 3e0
= 9
y = 2y + 3ex
⇒ y (x0) = 2(y0 ) + 3ex0
= 2(9) + 3e0
= 21
yiv
= 2y + 3ex
⇒ yiv
(x0) = 45
By Taylor’s series,
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

48. Taylor’s Series
Sol. Here,y = f(x, y) = 2y + 3ex
, x0 = 0 and y0 = 0.
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(0) + 3e0
= 3
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(3) + 3e0
= 9
y = 2y + 3ex
⇒ y (x0) = 2(y0 ) + 3ex0
= 2(9) + 3e0
= 21
yiv
= 2y + 3ex
⇒ yiv
(x0) = 45
By Taylor’s series,
y(x) = y0+(x−x0)y (x0)+
(x − x0)2
2!
y (x0)+
(x − x0)3
3!
y (x0)+. . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

49. Taylor’s Series
Sol. Here,y = f(x, y) = 2y + 3ex
, x0 = 0 and y0 = 0.
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(0) + 3e0
= 3
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(3) + 3e0
= 9
y = 2y + 3ex
⇒ y (x0) = 2(y0 ) + 3ex0
= 2(9) + 3e0
= 21
yiv
= 2y + 3ex
⇒ yiv
(x0) = 45
By Taylor’s series,
y(x) = y0+(x−x0)y (x0)+
(x − x0)2
2!
y (x0)+
(x − x0)3
3!
y (x0)+. . .
= 0 + x(3) +
x2
2!
(9) +
x3
3!
(21) +
x4
4!
(45) + . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

50. Taylor’s Series
Sol. Here,y = f(x, y) = 2y + 3ex
, x0 = 0 and y0 = 0.
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(0) + 3e0
= 3
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(3) + 3e0
= 9
y = 2y + 3ex
⇒ y (x0) = 2(y0 ) + 3ex0
= 2(9) + 3e0
= 21
yiv
= 2y + 3ex
⇒ yiv
(x0) = 45
By Taylor’s series,
y(x) = y0+(x−x0)y (x0)+
(x − x0)2
2!
y (x0)+
(x − x0)3
3!
y (x0)+. . .
= 0 + x(3) +
x2
2!
(9) +
x3
3!
(21) +
x4
4!
(45) + . . .
= 3x +
9x2
2
+
7x3
2
+
15x4
8
+ . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

51. Taylor’s Series
Sol. Here,y = f(x, y) = 2y + 3ex
, x0 = 0 and y0 = 0.
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(0) + 3e0
= 3
y = 2y + 3ex
⇒ y (x0) = 2(y0) + 3ex0
= 2(3) + 3e0
= 9
y = 2y + 3ex
⇒ y (x0) = 2(y0 ) + 3ex0
= 2(9) + 3e0
= 21
yiv
= 2y + 3ex
⇒ yiv
(x0) = 45
By Taylor’s series,
y(x) = y0+(x−x0)y (x0)+
(x − x0)2
2!
y (x0)+
(x − x0)3
3!
y (x0)+. . .
= 0 + x(3) +
x2
2!
(9) +
x3
3!
(21) +
x4
4!
(45) + . . .
= 3x +
9x2
2
+
7x3
2
+
15x4
8
+ . . .
y(0.2) =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

52. Taylor’s Series Method
Ex:
Use Taylor’s series method to solve
dy
dx
= x2
+ y2
,
y(0) = 1. Find y(0.1) correct up to 4 decimal
places.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

53. Taylor’s Series Method
Ex:
Use Taylor’s series method to solve
dy
dx
= x2
y − 1,
y(0) = 1. Find y(0.03).
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

54. Picard’s Method
Picard’s Method:
Consider the first order differential equation.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

55. Picard’s Method
Picard’s Method:
Consider the first order differential equation.
dy
dx
= f(x, y) − − − (1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

56. Picard’s Method
Picard’s Method:
Consider the first order differential equation.
dy
dx
= f(x, y) − − − (1)
subject to y(x0) = y0
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

57. Picard’s Method
Picard’s Method:
Consider the first order differential equation.
dy
dx
= f(x, y) − − − (1)
subject to y(x0) = y0
The equation (1) can be written as
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

58. Picard’s Method
Picard’s Method:
Consider the first order differential equation.
dy
dx
= f(x, y) − − − (1)
subject to y(x0) = y0
The equation (1) can be written as
dy = f(x, y)dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

59. Picard’s Method
Picard’s Method:
Consider the first order differential equation.
dy
dx
= f(x, y) − − − (1)
subject to y(x0) = y0
The equation (1) can be written as
dy = f(x, y)dx
Integrating between the limits for x and y, we get
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

60. Picard’s Method
Picard’s Method:
Consider the first order differential equation.
dy
dx
= f(x, y) − − − (1)
subject to y(x0) = y0
The equation (1) can be written as
dy = f(x, y)dx
Integrating between the limits for x and y, we get
y
y0
dy =
x
x0
f(x, y)dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

61. Picard’s Method
Picard’s Method:
Consider the first order differential equation.
dy
dx
= f(x, y) − − − (1)
subject to y(x0) = y0
The equation (1) can be written as
dy = f(x, y)dx
Integrating between the limits for x and y, we get
y
y0
dy =
x
x0
f(x, y)dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

62. Picard’s Method
(y − y0) =
x
x0
f(x, y)dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

63. Picard’s Method
(y − y0) =
x
x0
f(x, y)dx
y = y0 +
x
x0
f(x, y)dx − − − (2)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

64. Picard’s Method
(y − y0) =
x
x0
f(x, y)dx
y = y0 +
x
x0
f(x, y)dx − − − (2)
Equation (2) is known as integral equation and
can be solved by successive approximation or
iteration.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

65. Picard’s Method
(y − y0) =
x
x0
f(x, y)dx
y = y0 +
x
x0
f(x, y)dx − − − (2)
Equation (2) is known as integral equation and
can be solved by successive approximation or
iteration.
Now by Picard’s method, for 1st
approximation
y1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

66. Picard’s Method
(y − y0) =
x
x0
f(x, y)dx
y = y0 +
x
x0
f(x, y)dx − − − (2)
Equation (2) is known as integral equation and
can be solved by successive approximation or
iteration.
Now by Picard’s method, for 1st
approximation
y1
we replace y by y0 in f(x, y) in R.H.S of eq. (2),
we get
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

67. Picard’s Method
(y − y0) =
x
x0
f(x, y)dx
y = y0 +
x
x0
f(x, y)dx − − − (2)
Equation (2) is known as integral equation and
can be solved by successive approximation or
iteration.
Now by Picard’s method, for 1st
approximation
y1
we replace y by y0 in f(x, y) in R.H.S of eq. (2),
we get
y1 = y0 +
x
x0
f(x, y0)dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

68. Picard’s Method
For 2nd
approximation y2,
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

69. Picard’s Method
For 2nd
approximation y2,
we replace y by y1 in f(x, y) in R.H.S of eq. (2),
we get
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

70. Picard’s Method
For 2nd
approximation y2,
we replace y by y1 in f(x, y) in R.H.S of eq. (2),
we get
y2 = y0 +
x
x0
f(x, y1)dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

71. Picard’s Method
For 2nd
approximation y2,
we replace y by y1 in f(x, y) in R.H.S of eq. (2),
we get
y2 = y0 +
x
x0
f(x, y1)dx
In general,
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

72. Picard’s Method
For 2nd
approximation y2,
we replace y by y1 in f(x, y) in R.H.S of eq. (2),
we get
y2 = y0 +
x
x0
f(x, y1)dx
In general,
yn+1 = y0 +
x
x0
f(x, yn)dx for n = 0, 1, 2, . . .
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

73. Picard’s Method
For 2nd
approximation y2,
we replace y by y1 in f(x, y) in R.H.S of eq. (2),
we get
y2 = y0 +
x
x0
f(x, y1)dx
In general,
yn+1 = y0 +
x
x0
f(x, yn)dx for n = 0, 1, 2, . . .
stop the process when the two consecutive values
of y are same up to desired accuracy.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

74. Picard’s Method
Note:
This method is applicable to a limited class of
equations in which the successive integration can
be performed easily.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

75. Picard’s Method
Ex Using Picard’s method solve
dy
dx
= 3 + 2xy where y(0) = 1 for x = 0.1.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

76. Picard’s Method
Sol.:
By Picard’s method yn+1 = y0 +
x
x0
f(x, yn) dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

77. Picard’s Method
Sol.:
By Picard’s method yn+1 = y0 +
x
x0
f(x, yn) dx
Here x0 = 0, y0 = 1, f(x, y) = 3 + 2xy
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

78. Picard’s Method
Sol.:
By Picard’s method yn+1 = y0 +
x
x0
f(x, yn) dx
Here x0 = 0, y0 = 1, f(x, y) = 3 + 2xy
1st
approximation:
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

79. Picard’s Method
Sol.:
By Picard’s method yn+1 = y0 +
x
x0
f(x, yn) dx
Here x0 = 0, y0 = 1, f(x, y) = 3 + 2xy
1st
approximation:
put n = 0 and y0 = 1 in f(x, y)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

80. Picard’s Method
Sol.:
By Picard’s method yn+1 = y0 +
x
x0
f(x, yn) dx
Here x0 = 0, y0 = 1, f(x, y) = 3 + 2xy
1st
approximation:
put n = 0 and y0 = 1 in f(x, y)
y1 = 1 +
x
0
(3 + 2x) dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

81. Picard’s Method
Sol.:
By Picard’s method yn+1 = y0 +
x
x0
f(x, yn) dx
Here x0 = 0, y0 = 1, f(x, y) = 3 + 2xy
1st
approximation:
put n = 0 and y0 = 1 in f(x, y)
y1 = 1 +
x
0
(3 + 2x) dx
∴ y1 = 1 + 3x + x2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

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

83. Picard’s Method
2nd
approximation:
put n = 1 and y1 = 1 + 3x + x2
in f(x, y)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

84. Picard’s Method
2nd
approximation:
put n = 1 and y1 = 1 + 3x + x2
in f(x, y)
y2 = 1 +
x
0
3 + 2x(1 + 3x + x2
) dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

85. Picard’s Method
2nd
approximation:
put n = 1 and y1 = 1 + 3x + x2
in f(x, y)
y2 = 1 +
x
0
3 + 2x(1 + 3x + x2
) dx
= 1 +
x
0
3 + 2x + 6x2
+ 2x3
) dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

86. Picard’s Method
2nd
approximation:
put n = 1 and y1 = 1 + 3x + x2
in f(x, y)
y2 = 1 +
x
0
3 + 2x(1 + 3x + x2
) dx
= 1 +
x
0
3 + 2x + 6x2
+ 2x3
) dx
∴ y2 = 1 + 3x + x2
+ 2x3
+
x4
2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

87. Picard’s Method
2nd
approximation:
put n = 1 and y1 = 1 + 3x + x2
in f(x, y)
y2 = 1 +
x
0
3 + 2x(1 + 3x + x2
) dx
= 1 +
x
0
3 + 2x + 6x2
+ 2x3
) dx
∴ y2 = 1 + 3x + x2
+ 2x3
+
x4
2
which is approximate solution, putting x = 0.1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

88. Picard’s Method
2nd
approximation:
put n = 1 and y1 = 1 + 3x + x2
in f(x, y)
y2 = 1 +
x
0
3 + 2x(1 + 3x + x2
) dx
= 1 +
x
0
3 + 2x + 6x2
+ 2x3
) dx
∴ y2 = 1 + 3x + x2
+ 2x3
+
x4
2
which is approximate solution, putting x = 0.1
y(0.1) =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

89. Picard’s Method
2nd
approximation:
put n = 1 and y1 = 1 + 3x + x2
in f(x, y)
y2 = 1 +
x
0
3 + 2x(1 + 3x + x2
) dx
= 1 +
x
0
3 + 2x + 6x2
+ 2x3
) dx
∴ y2 = 1 + 3x + x2
+ 2x3
+
x4
2
which is approximate solution, putting x = 0.1
y(0.1) = 1.31205
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

90. Picard’s Method
Ex:
Using Picard’s method, obtain a solution upto
4th
approx of the equation
dy
dx
= y + x, y(0) = 1.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

91. Picard’s Method
Sol.:
By Picard’s method y = y0 +
x
x0
f(x, y) dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

92. Picard’s Method
Sol.:
By Picard’s method y = y0 +
x
x0
f(x, y) dx
Here x0 = 0, y0 = 1, f(x, y) = x + y
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

93. Picard’s Method
Sol.:
By Picard’s method y = y0 +
x
x0
f(x, y) dx
Here x0 = 0, y0 = 1, f(x, y) = x + y
1st
approximation: put y = y0 = 1 in f(x, y)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

94. Picard’s Method
Sol.:
By Picard’s method y = y0 +
x
x0
f(x, y) dx
Here x0 = 0, y0 = 1, f(x, y) = x + y
1st
approximation: put y = y0 = 1 in f(x, y)
‘y1 = 1 +
x
0
(1 + x) dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

95. Picard’s Method
Sol.:
By Picard’s method y = y0 +
x
x0
f(x, y) dx
Here x0 = 0, y0 = 1, f(x, y) = x + y
1st
approximation: put y = y0 = 1 in f(x, y)
‘y1 = 1 +
x
0
(1 + x) dx
∴ y1 = 1 + x +
x2
2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

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

97. Picard’s Method
2nd
approximation:
put y = y1 = 1 + x +
x2
2
in f(x, y)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

98. Picard’s Method
2nd
approximation:
put y = y1 = 1 + x +
x2
2
in f(x, y)
‘y2 = 1 +
x
0
1 + 2x +
x2
2
dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

99. Picard’s Method
2nd
approximation:
put y = y1 = 1 + x +
x2
2
in f(x, y)
‘y2 = 1 +
x
0
1 + 2x +
x2
2
dx
∴ y2 = 1 + x + x2
+
x3
6
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

100. Picard’s Method
3rd
approximation:
put y = 1 + x + x2
+
x3
6
in f(x, y)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

101. Picard’s Method
3rd
approximation:
put y = 1 + x + x2
+
x3
6
in f(x, y)
‘y3 = 1 +
x
0
1 + 2x + x2
+
x3
6
dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

102. Picard’s Method
3rd
approximation:
put y = 1 + x + x2
+
x3
6
in f(x, y)
‘y3 = 1 +
x
0
1 + 2x + x2
+
x3
6
dx
∴ y3 = 1 + x + x2
+
x3
3
+
x4
24
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

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

104. Picard’s Method
4th
approximation:
put y = 1 + x + x2
+
x3
3
+
x4
24
in f(x, y)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

105. Picard’s Method
4th
approximation:
put y = 1 + x + x2
+
x3
3
+
x4
24
in f(x, y)
‘y4 = 1 +
x
0
1 + 2x + x2
+
x3
3
+
x4
24
dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

106. Picard’s Method
4th
approximation:
put y = 1 + x + x2
+
x3
3
+
x4
24
in f(x, y)
‘y4 = 1 +
x
0
1 + 2x + x2
+
x3
3
+
x4
24
dx
∴ y4 = 1 + x + x2
+
x3
3
+
x4
12
+
x5
120
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

107. Picard’s Method
Ex:
Using Picard’s 2nd
approx. solution of the initial
value problem
dy
dx
= x2
+ y2
,for x = 0.4 correct to
4 decimal places given that y(0) = 0.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

108. Picard’s Method
Sol.:
By Picard’s method y = y0 +
x
x0
f(x, y) dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

109. Picard’s Method
Sol.:
By Picard’s method y = y0 +
x
x0
f(x, y) dx
Here x0 = 0, y0 = 0, f(x, y) = x2
+ y2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

110. Picard’s Method
Sol.:
By Picard’s method y = y0 +
x
x0
f(x, y) dx
Here x0 = 0, y0 = 0, f(x, y) = x2
+ y2
1st
approximation: put y = y0 = 0 in f(x, y)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

111. Picard’s Method
Sol.:
By Picard’s method y = y0 +
x
x0
f(x, y) dx
Here x0 = 0, y0 = 0, f(x, y) = x2
+ y2
1st
approximation: put y = y0 = 0 in f(x, y)
‘y1 = 0 +
x
0
(x2
+ 0) dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

112. Picard’s Method
Sol.:
By Picard’s method y = y0 +
x
x0
f(x, y) dx
Here x0 = 0, y0 = 0, f(x, y) = x2
+ y2
1st
approximation: put y = y0 = 0 in f(x, y)
‘y1 = 0 +
x
0
(x2
+ 0) dx
∴ y1 =
x3
3
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

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

114. Picard’s Method
2nd
approximation:
put y = y1 =
x3
3
in f(x, y)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

115. Picard’s Method
2nd
approximation:
put y = y1 =
x3
3
in f(x, y)
‘y2 = y0 +
x
0
x2
+
x3
3
2
dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

116. Picard’s Method
2nd
approximation:
put y = y1 =
x3
3
in f(x, y)
‘y2 = y0 +
x
0
x2
+
x3
3
2
dx
∴ y2 =
x3
3
+
x7
63
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

117. Picard’s Method
2nd
approximation:
put y = y1 =
x3
3
in f(x, y)
‘y2 = y0 +
x
0
x2
+
x3
3
2
dx
∴ y2 =
x3
3
+
x7
63
y(0.4) =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

118. Picard’s Method
Ex:
Find the value of y for x = 0.1 by Picard’s
method given that
dy
dx
=
y − x
y + x
,y(0) = 1.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

119. Picard’s Method
Sol.:
By Picard’s method y = y0 +
x
x0
f(x, y) dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

120. Picard’s Method
Sol.:
By Picard’s method y = y0 +
x
x0
f(x, y) dx
Here x0 = 0, y0 = 1, f(x, y) =
y − x
y + x
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

121. Picard’s Method
Sol.:
By Picard’s method y = y0 +
x
x0
f(x, y) dx
Here x0 = 0, y0 = 1, f(x, y) =
y − x
y + x
1st
approximation: put y = y0 = 1 in f(x, y)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

122. Picard’s Method
Sol.:
By Picard’s method y = y0 +
x
x0
f(x, y) dx
Here x0 = 0, y0 = 1, f(x, y) =
y − x
y + x
1st
approximation: put y = y0 = 1 in f(x, y)
y1 = 1 +
x
0
1 − x
1 + x
dx = 1 +
x
0
2 − (1 + x)
1 + x
dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

123. Picard’s Method
Sol.:
By Picard’s method y = y0 +
x
x0
f(x, y) dx
Here x0 = 0, y0 = 1, f(x, y) =
y − x
y + x
1st
approximation: put y = y0 = 1 in f(x, y)
y1 = 1 +
x
0
1 − x
1 + x
dx = 1 +
x
0
2 − (1 + x)
1 + x
dx
= 1 +
x
0
2
1 + x
− 1 dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

124. Picard’s Method
Sol.:
By Picard’s method y = y0 +
x
x0
f(x, y) dx
Here x0 = 0, y0 = 1, f(x, y) =
y − x
y + x
1st
approximation: put y = y0 = 1 in f(x, y)
y1 = 1 +
x
0
1 − x
1 + x
dx = 1 +
x
0
2 − (1 + x)
1 + x
dx
= 1 +
x
0
2
1 + x
− 1 dx
∴ y1 = 1 − x + 2log(1 + x)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

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

126. Picard’s Method
2nd
approximation:
put y = y1 = 1 + 2log(1 + x) − x in f(x, y)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

127. Picard’s Method
2nd
approximation:
put y = y1 = 1 + 2log(1 + x) − x in f(x, y)
y2 = 1 +
x
0
1 − x + 2log(1 + x) − x
1 − x + 2log(1 + x) + x
dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

128. Picard’s Method
2nd
approximation:
put y = y1 = 1 + 2log(1 + x) − x in f(x, y)
y2 = 1 +
x
0
1 − x + 2log(1 + x) − x
1 − x + 2log(1 + x) + x
dx
= 1 +
x
0
1 − 2x + 2log(1 + x)
1 + 2log(1 + x)
dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

129. Picard’s Method
2nd
approximation:
put y = y1 = 1 + 2log(1 + x) − x in f(x, y)
y2 = 1 +
x
0
1 − x + 2log(1 + x) − x
1 − x + 2log(1 + x) + x
dx
= 1 +
x
0
1 − 2x + 2log(1 + x)
1 + 2log(1 + x)
dx
= 1 +
x
0
1 −
2x
1 + 2log(1 + x)
dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

130. Picard’s Method
= 1 + x −
x
0
2x
1 + 2log(1 + x)
dx
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

131. Picard’s Method
= 1 + x −
x
0
2x
1 + 2log(1 + x)
dx
which is difficult to integrate therefore using 1st
approximation.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -

132. Picard’s Method
= 1 + x −
x
0
2x
1 + 2log(1 + x)
dx
which is difficult to integrate therefore using 1st
approximation.
y(0.1) =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations -