dy/dx - = 1 - y,y(0) = 0 is an example of

[cyberian]

in newton-cotes formula for finding the definite of a tabular function, which of the following taken as an approximate function then find the desire integral?

[cyberian]

@cyberian said in MTH603 Quiz 3 Solution and Discussion:

in newton-cotes formula for finding the definite of a tabular function, which of the following taken as an approximate function then find the desire integral?

In the Newton-Cotes formulas for numerical integration, a polynomial function is typically used as the approximate function to estimate the integral of a given tabular function. The Newton-Cotes formulas include several specific methods, such as the Trapezoidal Rule, Simpson’s Rule, and higher-order polynomial approximations.

Here are some of the commonly used Newton-Cotes formulas:

Trapezoidal Rule:
Approximates the function as a first-degree polynomial (a straight line) between each pair of points.

Simpson’s Rule:
Approximates the function as a second-degree polynomial (a parabola) over pairs of intervals.

Simpson’s 3/8 Rule:
Uses a cubic polynomial approximation over three subintervals.

Higher-Order Newton-Cotes Formulas:
Use higher-degree polynomials for approximation over more points.

To find the desired integral using one of these formulas, you would follow these general steps:

Select the appropriate Newton-Cotes formula:
Choose the formula based on the degree of accuracy you need and the number of data points available.

Divide the interval into subintervals:
Split the interval
[
𝑎
,
𝑏
]
[a,b] into
𝑛
n equal subintervals, where
𝑛
n depends on the chosen formula.

Apply the formula:
Use the selected Newton-Cotes formula to compute the integral. Here are the basic forms for the Trapezoidal Rule and Simpson’s

[cyberian]

in Simpson’s 1/3 rule, the global error is of …?

[cyberian]

@cyberian said in MTH603 Quiz 3 Solution and Discussion:

in Simpson’s 1/3 rule, the global error is of …?

In Simpson’s 1/3 rule for numerical integration, the global error (also known as the total error or the error over the entire interval) is of order
𝑂(ℎ4)

[cyberian]

givren that dy/dx= y-t/y+1 with the intial condition y=1, t=0 using euler’s method y at h=0.01; the value of y(0.01) is?

[cyberian]

@cyberian said in MTH603 Quiz 3 Solution and Discussion:

givren that dy/dx= y-t/y+1 with the intial condition y=1, t=0 using euler’s method y at h=0.01; the value of y(0.01) is?

To solve the differential equation
𝑑
𝑦
𝑑
𝑡

𝑦
−
𝑡
𝑦
+
1
dt
dy
​

y+1
y−t
​
with the initial condition
𝑦
(
0
)

1
y(0)=1 using Euler’s method with a step size
ℎ

0.01
h=0.01, we can follow these steps:

Identify the differential equation and initial condition:

𝑑
𝑦
𝑑
𝑡

𝑦
−
𝑡
𝑦
+
1
dt
dy
​

y+1
y−t
​

𝑦
(
0
)

1
y(0)=1
Euler’s method formula:

𝑦
𝑛
+
1

𝑦
𝑛
+
ℎ
⋅
𝑓
(
𝑡
𝑛
,
𝑦
𝑛
)
y
n+1
​
=y
n
​
+h⋅f(t
n
​
,y
n
​
)
where
𝑓
(
𝑡
,
𝑦
)

𝑦
−
𝑡
𝑦
+
1
f(t,y)=
y+1
y−t
​
.

Initial values:

𝑡
0

0
,
𝑦
0

1
t
0
​
=0,y
0
​
=1
Calculate
𝑦
1
y
1
​
using Euler’s method with
ℎ

0.01
h=0.01:

𝑡
1

𝑡
0
+
ℎ

0
+
0.01

0.01
t
1
​
=t
0
​
+h=0+0.01=0.01

𝑦
1

𝑦
0
+
ℎ
⋅
𝑓
(
𝑡
0
,
𝑦
0
)

1
+
0.01
⋅
1
−
0
1
+
1

1
+
0.01
⋅
1
2

1
+
0.01
⋅
0.5

1
+
0.005

1.005
y
1
​
=y
0
​
+h⋅f(t
0
​
,y
0
​
)=1+0.01⋅
1+1
1−0
​
=1+0.01⋅
2
1
​
=1+0.01⋅0.5=1+0.005=1.005
Therefore, the value of
𝑦
(
0.01
)
y(0.01) using Euler’s method with a step size of
ℎ

0.01
h=0.01 is approximately
1.005
1.005.

[cyberian]

ther percentahe error in numerial integration s defined as?

[cyberian]

@cyberian said in MTH603 Quiz 3 Solution and Discussion:

ther percentahe error in numerial integration s defined as?

The percentage error in numerical integration is defined as the relative difference between the exact value of the integral and the approximate value obtained using a numerical method. It is given by the formula:

[cyberian]

if yn+1 = yn +1/6 (k1+2k2+2k3+k4) then k2 is

[cyberian]

@cyberian said in MTH603 Quiz 3 Solution and Discussion:

if yn+1 = yn +1/6 (k1+2k2+2k3+k4) then k2 is

The formula you provided is related to the Runge-Kutta method, specifically the classical fourth-order Runge-Kutta method (RK4). The formula for

[cyberian]

in solving the follwing differnital equation
y =x+y; y(0)=1
h=0.2

[cyberian]

@cyberian said in MTH603 Quiz 3 Solution and Discussion:

in solving the follwing differnital equation
y =x+y; y(0)=1
h=0.2

To solve the differential equation

𝑦
′
=
𝑥
+
𝑦
y 
′

=x+y with the initial condition

𝑦
(
0
)
=
1

y(0)=1 using the step size

ℎ
=
0.2

h=0.2, we can use the Euler method. Here are the steps:

[cyberian]

which of the following points the max value of 2nd deruvative of function f(x)=-(2/x) in the inteval : [1,4] exits

[cyberian]

@cyberian said in MTH603 Quiz 3 Solution and Discussion:

which of the following points the max value of 2nd deruvative of function f(x)=-(2/x) in the inteval : [1,4] exits

To find where the maximum value of the second derivative of the function

𝑓
(
𝑥
)
=
−
2
𝑥
f(x)=− 
x
2


[
1
,
4
]

[1,4], we will first find the second derivative of the function and then determine where it achieves its maximum value within the given interval.

[zaasmi]

@cyberian said in MTH603 Quiz 3 Solution and Discussion:

which of the following reason lead towards the numerical integration methods?

Analytical evaluation of integral is very complicated
All above choices are true
Integrand is given in tabular form
Analytical evaluation of integral is impossible

The reasons that lead towards the use of numerical integration methods include:

Analytical evaluation of integral is very complicated
Integrand is given in tabular form
Analytical evaluation of integral is impossible
Therefore, the correct answer is: All above choices are true

[zaasmi]

In Simpson’s 3/8 rule, the global error is of ………………

Select correct option:
O(h2)
O(h3)
O(h4)
None of the given choices