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Numerical MethodsOrdinary Differential Equations - 2
Dr. N. B. Vyas
Department of Mathematics,Atmiya Institute of Tech. and Science,
Rajkot (Gujarat) - [email protected]
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Ordinary Differential Equations
Euler’s Method:
Consider the 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) + . . . - - - (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 - 2
Ordinary Differential Equations
Euler’s Method:
Consider the 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) + . . . - - - (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 - 2
Ordinary Differential Equations
Euler’s Method:
Consider the 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) + . . . - - - (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 - 2
Ordinary Differential Equations
Euler’s Method:
Consider the 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) + . . . - - - (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 - 2
Euler’s Method
If h is chosen small enough then we may neglect the second andhigher order term of h.
y1 = y0 + hf(x0, y0)Which is Euler’s first approximation.The general step for Euler method isyi+1 = yi + hf(xi, yi) where i = 0, 1, 2....
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Euler’s Method
If h is chosen small enough then we may neglect the second andhigher order term of h.y1 = y0 + hf(x0, y0)
Which is Euler’s first approximation.The general step for Euler method isyi+1 = yi + hf(xi, yi) where i = 0, 1, 2....
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Euler’s Method
If h is chosen small enough then we may neglect the second andhigher order term of h.y1 = y0 + hf(x0, y0)Which is Euler’s first approximation.
The general step for Euler method isyi+1 = yi + hf(xi, yi) where i = 0, 1, 2....
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Euler’s Method
If h is chosen small enough then we may neglect the second andhigher 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 - 2
Euler’s Method
If h is chosen small enough then we may neglect the second andhigher order term of h.y1 = y0 + hf(x0, y0)Which is Euler’s first approximation.The general step for Euler method isyi+1 = yi + hf(xi, yi) where i = 0, 1, 2....
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Euler’s Method
Ex.: Use Euler’s method to find y(1.6) given thatdy
dx= xy
12 , y(1) = 1
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 anddy
dx= f(x, y) = xy
12
we take h = 0.2
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.2)(1)(1)12
= 1.2
x1 = x0 + h = 1 + 0.2 = 1.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 anddy
dx= f(x, y) = xy
12
we take h = 0.2
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.2)(1)(1)12
= 1.2
x1 = x0 + h = 1 + 0.2 = 1.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 anddy
dx= f(x, y) = xy
12
we take h = 0.2
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.2)(1)(1)12
= 1.2
x1 = x0 + h = 1 + 0.2 = 1.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 anddy
dx= f(x, y) = xy
12
we take h = 0.2
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.2)(1)(1)12
= 1.2
x1 = x0 + h = 1 + 0.2 = 1.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 anddy
dx= f(x, y) = xy
12
we take h = 0.2
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.2)(1)(1)12
= 1.2
x1 = x0 + h = 1 + 0.2 = 1.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Euler’s Method
Sol.:
Here x0 = 1, y0 = 1 anddy
dx= f(x, y) = xy
12
we take h = 0.2
1st approximation:
y1 = y0 + hf(x0, y0)
= 1 + (0.2)(1)(1)12
= 1.2
x1 = x0 + h = 1 + 0.2 = 1.2
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 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)12
= 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 - 2
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 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)12
= 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 - 2
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 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)12
= 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 - 2
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 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)12
= 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 - 2
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 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)12
= 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 - 2
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 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)12
= 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 - 2
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 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)12
= 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 - 2
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 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)12
= 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 - 2
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 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)12
= 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 - 2
Euler’s Method
2nd approximation:
y2 = y1 + hf(x1, y1)
= 1.2 + (0.2)(1.2)(1.2)12
= 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)12
= 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 - 2
Euler’s Method
Ex.: Using Euler’s method, find y(0.2), givendy
dx= y − 2x
y, y(0) = 1. (Take h = 0.1)
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 anddy
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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 anddy
dx= f(x, y) = y − 2x
ywe 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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 anddy
dx= f(x, y) = y − 2x
ywe 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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 anddy
dx= f(x, y) = y − 2x
ywe 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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 anddy
dx= f(x, y) = y − 2x
ywe 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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 anddy
dx= f(x, y) = y − 2x
ywe 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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1 anddy
dx= f(x, y) = y − 2x
ywe 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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
Euler’s Method
Ex.: Use Euler’s method to obtain an approx value
of y(0.4) for the equationdy
dx= x + y, y(0) = 1 with
h = 0.1.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 anddy
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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 anddy
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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 anddy
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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 anddy
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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 anddy
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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.1 anddy
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
Euler’s Method
Ex.: Givendy
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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 anddy
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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 anddy
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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 anddy
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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 anddy
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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 anddy
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 - 2
Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.02 anddy
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
Euler’s Method
Ex.: Find y(2) fordy
dx=
y
x, y(1) = 1.
using Euler’s method, take h = 0.2.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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 valuesof y agree.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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 valuesof y agree.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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 valuesof y agree.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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 valuesof y agree.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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 valuesof y agree.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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 valuesof y agree.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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 valuesof y agree.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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 wayas above.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,we find y2y2 = 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 wayas above.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,we find y2y2 = 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 wayas above.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,we find y2y2 = 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 wayas above.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,we find y2y2 = 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 wayas above.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,we find y2y2 = 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 wayas above.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
Once y1 is obtained to desired degree of accuracy,we find y2y2 = 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 wayas above.
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
Ex.: Solvedy
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 - 2
Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.05 anddy
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 - 2
Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.05 anddy
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 - 2
Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.05 anddy
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 - 2
Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.05 anddy
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 - 2
Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 1, h = 0.05 anddy
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 - 2
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 1 + 0.052 [(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 + 0.052 [(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 - 2
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 1 + 0.052 [(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 + 0.052 [(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 - 2
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 1 + 0.052 [(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 + 0.052 [(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 - 2
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 1 + 0.052 [(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 + 0.052 [(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 - 2
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 1 + 0.052 [(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 + 0.052 [(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 - 2
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 1 + 0.052 [(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 + 0.052 [(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 - 2
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 1 + 0.052 [(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 + 0.052 [(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 - 2
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 1 + 0.052 [(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 + 0.052 [(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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 0, h = 0.2 anddy
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 - 2
Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 0, h = 0.2 anddy
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 - 2
Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 0, h = 0.2 anddy
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 - 2
Modified Euler’s Method
Sol.:
Here x0 = 0, y0 = 0, h = 0.2 anddy
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 - 2
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 0 + 0.22 [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 + 0.22 [f(0, 0) + f(0.2, 0.24214)] = 0.24635
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 0 + 0.22 [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 + 0.22 [f(0, 0) + f(0.2, 0.24214)] = 0.24635
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 0 + 0.22 [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 + 0.22 [f(0, 0) + f(0.2, 0.24214)] = 0.24635
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 0 + 0.22 [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 + 0.22 [f(0, 0) + f(0.2, 0.24214)] = 0.24635
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 0 + 0.22 [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 + 0.22 [f(0, 0) + f(0.2, 0.24214)] = 0.24635
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 0 + 0.22 [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 + 0.22 [f(0, 0) + f(0.2, 0.24214)] =
0.24635
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
1st approximation:
y(1)1 = y0 +
h
2[f(x0, y0) + f(x1, y1)]
= 0 + 0.22 [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 + 0.22 [f(0, 0) + f(0.2, 0.24214)] = 0.24635
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
3rd approximation:
y(3)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
= 0 + 0.22 [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 + 0.22 [f(0, 0) + f(0.2, 0.24678)] = 0.24681
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
3rd approximation:
y(3)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
= 0 + 0.22 [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 + 0.22 [f(0, 0) + f(0.2, 0.24678)] = 0.24681
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
3rd approximation:
y(3)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
= 0 + 0.22 [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 + 0.22 [f(0, 0) + f(0.2, 0.24678)] = 0.24681
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
3rd approximation:
y(3)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
= 0 + 0.22 [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 + 0.22 [f(0, 0) + f(0.2, 0.24678)] = 0.24681
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
3rd approximation:
y(3)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
= 0 + 0.22 [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 + 0.22 [f(0, 0) + f(0.2, 0.24678)] = 0.24681
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
3rd approximation:
y(3)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
= 0 + 0.22 [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 + 0.22 [f(0, 0) + f(0.2, 0.24678)] =
0.24681
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
Modified Euler’s Method
3rd approximation:
y(3)1 = y0 +
h
2
[f(x0, y0) + f(x1, y
(2)1 )]
= 0 + 0.22 [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 + 0.22 [f(0, 0) + f(0.2, 0.24678)] = 0.24681
Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2
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 - 2