Use Euler's method with h=0.1 to find approximate values?

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Homework Help Overview

The discussion revolves around using Euler's method with a step size of h=0.1 to approximate the solution of the initial-value problem defined by the differential equation y'+3y=7e^(4x) with the initial condition y(0)=2. Participants are tasked with finding approximate values at specified points from x=0 to x=1.0.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to apply Euler's method and presents initial calculations for y(0.1) and y(0.2). Some participants suggest using a table format to organize results. Questions arise regarding the necessity of calculating y(1) and the interpretation of the requested values from y(1) to y(10).

Discussion Status

Participants have provided feedback on the original poster's calculations, indicating that the approach seems correct so far. There is an ongoing exploration of how to present the results effectively, with some participants expressing uncertainty about their preferred format. The discussion is active, with clarifications being sought and offered.

Contextual Notes

There are indications of confusion regarding the need to calculate y(1) and how to interpret the results in terms of the requested values. The use of a table format is suggested as a means to clarify the calculations, but no consensus on the format has been reached.

Math10
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Homework Statement


Use Euler's method with h=0.1 to find approximate values of the solution of the initial-value problem y'+3y=7e^(4x), y(0)=2 at x=0, 0.1, 0.2, 0.3, ..., 1.0.

Homework Equations


f(x, y)=7e^(4x)-3y
x0=0, y0=2

The Attempt at a Solution


y(0.1)=y1=y0+f(0, 2)(0.1)=2+(0,1)f(0, 2)=2.1
y(0.2)=y2=y1+f(0.1, 2.1)(0.1)=2.1+(0.1)f(0.1, 2.1)=2.51428
And so on...
But am I doing it right? If not, then please correct me.
 
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Looks good so far.

You could use a table format like this to show the rest of your results:

$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Step }n & y_n & x_n & f(x_n,y_n) & h & \Delta y & y_{n+1} \\
\hline
0 & 2 & 0 & 1 & 0.1 & 0.1 & 2.1 \\
\hline
1 & 2.1 & 0.1 & 4.1428 & 0.1 & 0.41428 & 2.51428 \\
\hline
2 & 2.51428 & 0.2 & ?? & 0.1 & ? & ?? \\
\hline
3 & ?? & 0.3 & ?? & 0.1 & ? & ?? \\
\hline
4 & ?? & 0.4 & ?? & 0.1 & ? & ?? \\
\hline
5 & ?? & 0.5 & ?? & 0.1 & ? & ?? \\
\hline
6 & ?? & 0.6 & ?? & 0.1 & ? & ?? \\
\hline
7 & ?? & 0.7 & ?? & 0.1 & ? & ?? \\
\hline
8 & ?? & 0.8 & ?? & 0.1 & ? & ?? \\
\hline
9 & ?? & 0.9 & ?? & 0.1 & ? & ?? \\
\hline
\end{array}
$$
 
Do I nee to solve for y(1)? Because it says x=0, 0.1, 0.2, 0.3, ..., 1.0.
 
Math10 said:
Do I nee to solve for y(1)? Because it says x=0, 0.1, 0.2, 0.3, ..., 1.0.
Yes. I don't know why you would think you don't need to.

y(1) means the y value when x = 1.0.
 
Joffan said:
Looks good so far.

You could use a table format like this to show the rest of your results:

$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Step }n & y_n & x_n & f(x_n,y_n) & h & \Delta y & y_{n+1} \\
\hline
0 & 2 & 0 & 1 & 0.1 & 0.1 & 2.1 \\
\hline
1 & 2.1 & 0.1 & 4.1428 & 0.1 & 0.41428 & 2.51428 \\
\hline
2 & 2.51428 & 0.2 & ?? & 0.1 & ? & ?? \\
\hline
3 & ?? & 0.3 & ?? & 0.1 & ? & ?? \\
\hline
4 & ?? & 0.4 & ?? & 0.1 & ? & ?? \\
\hline
5 & ?? & 0.5 & ?? & 0.1 & ? & ?? \\
\hline
6 & ?? & 0.6 & ?? & 0.1 & ? & ?? \\
\hline
7 & ?? & 0.7 & ?? & 0.1 & ? & ?? \\
\hline
8 & ?? & 0.8 & ?? & 0.1 & ? & ?? \\
\hline
9 & ?? & 0.9 & ?? & 0.1 & ? & ?? \\
\hline
\end{array}
$$
PF now supports various table bbcode tags, an alternative to using LaTeX. Here are the header row and two rows of the first three columns of the table above.
Step (n)ynxn
[TD1]0[/TD1][TD1]2[/TD1] [TD1]0[/TD1] [TD1]1[/TD1][TD1]2.1[/TD1] [TD1]0.1[/TD1]
 
So the answer starts with y1 to y10, right?
 
Math10 said:
So the answer starts with y1 to y10, right?
Those are the numbers requested. Note that in the table I gave you, y10 is in the bottom right corner - this is our Euler-method estimate of y(1.0)
Mark44 said:
PF now supports various table bbcode tags, an alternative to using LaTeX. Here are the header row and two rows of the first three columns of the table above.
Step (n)ynxn
[TD1]0[/TD1][TD1]2[/TD1] [TD1]0[/TD1] [TD1]1[/TD1][TD1]2.1[/TD1] [TD1]0.1[/TD1]

Thanks... not sure which I prefer really, but the compact output is good.
 
Last edited:
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Joffan said:
Thanks... not sure which I prefer really, but the compact output is good.
I don't which I like better either. It's pretty much six of one, half a dozen of the other.
 

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