In Problems 1–4 construct a table comparing the indicated values of y(x) using Euler’s method, the improved Euler’s method, and the RK4 method. Compute to four rounded decimal places. First use h = 0.1 and then use h = 0.05.
The value of
Explanation of Solution
Given:
The linear differential equation is
Concept used:
Runga-Kutta method:
The solution of a linear differential equation of the form
Euler’s method
The approximated solution for the first order linear differential equation of the form
Improved Euler’s method:
As per the Improved Euler’s method the solution of a linear differential equation of the form
Calculation:
The linear differential equation is given as follows:
The value of step size is given as
The initial value is given as
The given differential equation is of the form
This implies that
Obtain the solution of the given differential equation by Euler’s method for
As per the Euler’s method the solution of a linear differential equation of the form
Substitute
Substitute the value of
Therefore, the value of
Similarly, use the above procedure and the value of
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Table
Table
Obtain the solution of the given differential equation by Improved Euler’s method for
As per the Improved Euler’s method the solution of a linear differential equation of the form
Substitute
Substitute the value of
Substitute
Substitute the value of
Therefore, the value of
Similarly, with above procedure used the value of
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Table
Table
Obtain the solution of the given differential equation by Euler’s modified method for
As per the Runga-Kutta method of fourth order the value of
Here, the value of
Substitute
Calculate the value of
Calculate the value of
Further solve the above equation.
Calculate the value of
Calculate the value of
Substitute the value of
Therefore, the value of
Similarly, with above procedure used the value of
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Table
Table
The table which shows a comparison between the values of
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Table
Thus, table
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Chapter 9 Solutions
First Course in Differential Equations (Instructor's)
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