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

[Math10]

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.

 

[Joffan]

Looks good so far.

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

$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Step }n & y_n & x_n & f(x_n,y_n) & h & \Delta y & y_{n+1} \\
\hline
0 & 2 & 0 & 1 & 0.1 & 0.1 & 2.1 \\
\hline
1 & 2.1 & 0.1 & 4.1428 & 0.1 & 0.41428 & 2.51428 \\
\hline
2 & 2.51428 & 0.2 & ?? & 0.1 & ? & ?? \\
\hline
3 & ?? & 0.3 & ?? & 0.1 & ? & ?? \\
\hline
4 & ?? & 0.4 & ?? & 0.1 & ? & ?? \\
\hline
5 & ?? & 0.5 & ?? & 0.1 & ? & ?? \\
\hline
6 & ?? & 0.6 & ?? & 0.1 & ? & ?? \\
\hline
7 & ?? & 0.7 & ?? & 0.1 & ? & ?? \\
\hline
8 & ?? & 0.8 & ?? & 0.1 & ? & ?? \\
\hline
9 & ?? & 0.9 & ?? & 0.1 & ? & ?? \\
\hline
\end{array}
$$

 

[Math10]

Do I nee to solve for y(1)? Because it says x=0, 0.1, 0.2, 0.3, ..., 1.0.

 

[Mark44]

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.

 

[Mark44]

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)	yn	xn

[TD1]0[/TD1][TD1]2[/TD1] [TD1]0[/TD1] [TD1]1[/TD1][TD1]2.1[/TD1] [TD1]0.1[/TD1]

 

[Math10]

So the answer starts with y₁ to y₁₀, right?

 

[Joffan]

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)

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)	yn	xn

[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.

 

[Mark44]

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.

 

[Math10]

Thank you, @Joffan .