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Numerical MethodsOrdinary Differential Equations - 1
Dr. N. B. Vyas
Department of Mathematics,Atmiya Institute of Technology & Science,
Rajkot (Gujarat) - [email protected]
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
Ordinary Differential Equations
Taylor’s Series Method:
Consider the first order Differential Equationdy
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 - 1
Ordinary Differential Equations
Taylor’s Series Method:
Consider the first order Differential Equationdy
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 - 1
Taylor’s Series Method
Ex:1
Solve y′ = x + y, y(0) = 1 by Taylor’s seriesmethod. Hence find values of y at x = 0.1 andx = 0.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
Taylor’s Series
Ex Using Taylor’s series method, obtain the solution
ofdy
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 - 1
Taylor’s Series
Sol. Here,y′ = f(x, y) = 3x + y2, x0 = 0 and y0 = 1.
y′ = 3x + y2 ⇒ y′(x0) = 3(x0) + y20 = 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 - 1
Taylor’s Series
Sol. Here,y′ = f(x, y) = 3x + y2, x0 = 0 and y0 = 1.
y′ = 3x + y2
⇒ y′(x0) = 3(x0) + y20 = 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 - 1
Taylor’s Series
Sol. Here,y′ = f(x, y) = 3x + y2, x0 = 0 and y0 = 1.
y′ = 3x + y2 ⇒ y′(x0) = 3(x0) + y20 = 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 - 1
Taylor’s Series
Sol. Here,y′ = f(x, y) = 3x + y2, x0 = 0 and y0 = 1.
y′ = 3x + y2 ⇒ y′(x0) = 3(x0) + y20 = 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 - 1
Taylor’s Series
Sol. Here,y′ = f(x, y) = 3x + y2, x0 = 0 and y0 = 1.
y′ = 3x + y2 ⇒ y′(x0) = 3(x0) + y20 = 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 - 1
Taylor’s Series
Sol. Here,y′ = f(x, y) = 3x + y2, x0 = 0 and y0 = 1.
y′ = 3x + y2 ⇒ y′(x0) = 3(x0) + y20 = 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 - 1
Taylor’s Series
Sol. Here,y′ = f(x, y) = 3x + y2, x0 = 0 and y0 = 1.
y′ = 3x + y2 ⇒ y′(x0) = 3(x0) + y20 = 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 - 1
Taylor’s Series
Sol. Here,y′ = f(x, y) = 3x + y2, x0 = 0 and y0 = 1.
y′ = 3x + y2 ⇒ y′(x0) = 3(x0) + y20 = 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 - 1
Taylor’s Series
Sol. Here,y′ = f(x, y) = 3x + y2, x0 = 0 and y0 = 1.
y′ = 3x + y2 ⇒ y′(x0) = 3(x0) + y20 = 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 - 1
Taylor’s Series
Sol. Here,y′ = f(x, y) = 3x + y2, x0 = 0 and y0 = 1.
y′ = 3x + y2 ⇒ y′(x0) = 3(x0) + y20 = 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 - 1
Taylor’s Series
Sol. Here,y′ = f(x, y) = 3x + y2, x0 = 0 and y0 = 1.
y′ = 3x + y2 ⇒ y′(x0) = 3(x0) + y20 = 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 - 1
Taylor’s Series
Sol. Here,y′ = f(x, y) = 3x + y2, x0 = 0 and y0 = 1.
y′ = 3x + y2 ⇒ y′(x0) = 3(x0) + y20 = 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 - 1
Taylor’s Series
Ex Using Taylor’s series method, find the solution ofdy
dx= 2y + 3ex, y(0) = 0,at x = 0.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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(y′0) + 3ex0 = 2(3) + 3e0 = 9
y′′′ = 2y′′ + 3ex ⇒ y′′′(x0) = 2(y′′0) + 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 - 1
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(y′0) + 3ex0 = 2(3) + 3e0 = 9
y′′′ = 2y′′ + 3ex ⇒ y′′′(x0) = 2(y′′0) + 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 - 1
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(y′0) + 3ex0 = 2(3) + 3e0 = 9
y′′′ = 2y′′ + 3ex ⇒ y′′′(x0) = 2(y′′0) + 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 - 1
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(y′0) + 3ex0 = 2(3) + 3e0 = 9
y′′′ = 2y′′ + 3ex ⇒ y′′′(x0) = 2(y′′0) + 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 - 1
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(y′0) + 3ex0 = 2(3) + 3e0 = 9
y′′′ = 2y′′ + 3ex ⇒ y′′′(x0) = 2(y′′0) + 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 - 1
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(y′0) + 3ex0 = 2(3) + 3e0 = 9
y′′′ = 2y′′ + 3ex
⇒ y′′′(x0) = 2(y′′0) + 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 - 1
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(y′0) + 3ex0 = 2(3) + 3e0 = 9
y′′′ = 2y′′ + 3ex ⇒ y′′′(x0) = 2(y′′0) + 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 - 1
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(y′0) + 3ex0 = 2(3) + 3e0 = 9
y′′′ = 2y′′ + 3ex ⇒ y′′′(x0) = 2(y′′0) + 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 - 1
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(y′0) + 3ex0 = 2(3) + 3e0 = 9
y′′′ = 2y′′ + 3ex ⇒ y′′′(x0) = 2(y′′0) + 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 - 1
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(y′0) + 3ex0 = 2(3) + 3e0 = 9
y′′′ = 2y′′ + 3ex ⇒ y′′′(x0) = 2(y′′0) + 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 - 1
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(y′0) + 3ex0 = 2(3) + 3e0 = 9
y′′′ = 2y′′ + 3ex ⇒ y′′′(x0) = 2(y′′0) + 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 - 1
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(y′0) + 3ex0 = 2(3) + 3e0 = 9
y′′′ = 2y′′ + 3ex ⇒ y′′′(x0) = 2(y′′0) + 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 - 1
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(y′0) + 3ex0 = 2(3) + 3e0 = 9
y′′′ = 2y′′ + 3ex ⇒ y′′′(x0) = 2(y′′0) + 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 - 1
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(y′0) + 3ex0 = 2(3) + 3e0 = 9
y′′′ = 2y′′ + 3ex ⇒ y′′′(x0) = 2(y′′0) + 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 - 1
Taylor’s Series Method
Ex:
Use Taylor’s series method to solvedy
dx= x2 + y2,
y(0) = 1. Find y(0.1) correct up to 4 decimalplaces.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
Taylor’s Series Method
Ex:
Use Taylor’s series method to solvedy
dx= x2y− 1,
y(0) = 1. Find y(0.03).
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 andcan be solved by successive approximation oriteration.
Now by Picard’s method, for 1st approximationy1we 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 - 1
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 andcan be solved by successive approximation oriteration.
Now by Picard’s method, for 1st approximationy1we 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 - 1
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 andcan be solved by successive approximation oriteration.
Now by Picard’s method, for 1st approximationy1we 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 - 1
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 andcan be solved by successive approximation oriteration.
Now by Picard’s method, for 1st approximationy1
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 - 1
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 andcan be solved by successive approximation oriteration.
Now by Picard’s method, for 1st approximationy1we 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 - 1
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 andcan be solved by successive approximation oriteration.
Now by Picard’s method, for 1st approximationy1we 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 - 1
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 valuesof y are same up to desired accuracy.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 valuesof y are same up to desired accuracy.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 valuesof y are same up to desired accuracy.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 valuesof y are same up to desired accuracy.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 valuesof y are same up to desired accuracy.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 valuesof y are same up to desired accuracy.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
Picard’s Method
Note:
This method is applicable to a limited class ofequations in which the successive integration canbe performed easily.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
Picard’s Method
Ex Using Picard’s method solvedy
dx= 3 + 2xy where y(0) = 1 for x = 0.1.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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
2which is approximate solution, putting x = 0.1
y(0.1) = 1.31205
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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
2which is approximate solution, putting x = 0.1
y(0.1) = 1.31205
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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
2which is approximate solution, putting x = 0.1
y(0.1) = 1.31205
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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
2which is approximate solution, putting x = 0.1
y(0.1) = 1.31205
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 - 1
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
2which is approximate solution, putting x = 0.1
y(0.1) = 1.31205
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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
2which is approximate solution, putting x = 0.1
y(0.1) =
1.31205
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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
2which is approximate solution, putting x = 0.1
y(0.1) = 1.31205
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
Picard’s Method
Ex:
Using Picard’s method, obtain a solution upto
4th approx of the equationdy
dx= y + x, y(0) = 1.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
Picard’s Method
2nd approximation:
put y = y1 = 1 + x +x2
2in 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 - 1
Picard’s Method
2nd approximation:
put y = y1 = 1 + x +x2
2in 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 - 1
Picard’s Method
2nd approximation:
put y = y1 = 1 + x +x2
2in 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 - 1
Picard’s Method
2nd approximation:
put y = y1 = 1 + x +x2
2in 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 - 1
Picard’s Method
3rd approximation:
put y = 1 + x + x2 +x3
6in 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 - 1
Picard’s Method
3rd approximation:
put y = 1 + x + x2 +x3
6in 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 - 1
Picard’s Method
3rd approximation:
put y = 1 + x + x2 +x3
6in 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 - 1
Picard’s Method
4th approximation:
put y = 1 + x + x2 +x3
3+
x4
24in 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 - 1
Picard’s Method
4th approximation:
put y = 1 + x + x2 +x3
3+
x4
24in 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 - 1
Picard’s Method
4th approximation:
put y = 1 + x + x2 +x3
3+
x4
24in 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 - 1
Picard’s Method
4th approximation:
put y = 1 + x + x2 +x3
3+
x4
24in 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 - 1
Picard’s Method
Ex:
Using Picard’s 2nd approx. solution of the initial
value problemdy
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
Picard’s Method
2nd approximation:
put y = y1 =x3
3in f(x, y)
‘y2 = y0 +
∫ x
0
[x2 +
(x3
3
)2]dx
∴ y2 =x3
3+
x7
63y(0.4) =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
Picard’s Method
2nd approximation:
put y = y1 =x3
3in f(x, y)
‘y2 = y0 +
∫ x
0
[x2 +
(x3
3
)2]dx
∴ y2 =x3
3+
x7
63y(0.4) =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
Picard’s Method
2nd approximation:
put y = y1 =x3
3in f(x, y)
‘y2 = y0 +
∫ x
0
[x2 +
(x3
3
)2]dx
∴ y2 =x3
3+
x7
63y(0.4) =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
Picard’s Method
2nd approximation:
put y = y1 =x3
3in 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 - 1
Picard’s Method
2nd approximation:
put y = y1 =x3
3in f(x, y)
‘y2 = y0 +
∫ x
0
[x2 +
(x3
3
)2]dx
∴ y2 =x3
3+
x7
63y(0.4) =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
Picard’s Method
Ex:
Find the value of y for x = 0.1 by Picard’s
method given thatdy
dx=
y − x
y + x,y(0) = 1.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 + xdx = 1 +
∫ x
0
2− (1 + x)
1 + xdx
= 1 +
∫ x
0
[2
1 + x− 1
]dx
∴ y1 = 1− x + 2log(1 + x)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 + xdx = 1 +
∫ x
0
2− (1 + x)
1 + xdx
= 1 +
∫ x
0
[2
1 + x− 1
]dx
∴ y1 = 1− x + 2log(1 + x)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 + xdx = 1 +
∫ x
0
2− (1 + x)
1 + xdx
= 1 +
∫ x
0
[2
1 + x− 1
]dx
∴ y1 = 1− x + 2log(1 + x)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 + xdx = 1 +
∫ x
0
2− (1 + x)
1 + xdx
= 1 +
∫ x
0
[2
1 + x− 1
]dx
∴ y1 = 1− x + 2log(1 + x)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 + xdx = 1 +
∫ x
0
2− (1 + x)
1 + xdx
= 1 +
∫ x
0
[2
1 + x− 1
]dx
∴ y1 = 1− x + 2log(1 + x)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 + xdx = 1 +
∫ x
0
2− (1 + x)
1 + xdx
= 1 +
∫ x
0
[2
1 + x− 1
]dx
∴ y1 = 1− x + 2log(1 + x)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
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 - 1
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 - 1
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 - 1
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 - 1
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 - 1
Picard’s Method
= 1 + x−∫ x
0
2x
1 + 2log(1 + x)dx
which is difficult to integrate therefore using 1stapproximation.
y(0.1) =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1
Picard’s Method
= 1 + x−∫ x
0
2x
1 + 2log(1 + x)dx
which is difficult to integrate therefore using 1stapproximation.
y(0.1) =
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 1