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Concept explainers
(a)
Program Description: Purpose of the problem is to obtain the approximate values of
(a)
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Explanation of Solution
Given information:
The differential problem is
The exact solution of the differential equation is
The two step size is
Calculation:
The differential problem can be represented as,
The Euler’s formula for
The Euler’s formula for
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 1 for
Substitute 1 for
Substitute
Substitute
Therefore, the value of
Conclusion:
Thus, the approximate values of
(b)
Program Description: Purpose of the problem is to find the approximate values of
(b)
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Explanation of Solution
Given information:
The differential problem is
The exact solution of the differential equation is
The two step size is
Calculation:
The improved Euler’s formula for predicators
The improved Euler’s formula for predicators
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 0 for
The improved Euler’s formula for correctors is shown below.
The improved Euler’s formula for correctors
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 0 for
Therefore, the value of
Conclusion:
Thus, the approximate values of
(c)
Program Description: Purpose of the problem is to find the approximate values of
(c)
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Explanation of Solution
Given information:
The differential problem is
The exact solution of the differential equation is
The two step size is
Calculation:
The Runge-Kutta iteration formulas are shown below.
The Runge-Kutta iteration formulas are shown below
The value of
The value of
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute 0 for
Substitute
Substitute
The exact solution
The exact solution
It can be observed that the Runge-Kutta method is more closes to the exact solution of the differential equation.
Conclusion:
Thus, the approximate values of
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Chapter 4 Solutions
Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis
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