Yes. I don't know why you would think you don't need to.Math10 said:Do I nee to solve for y(1)? Because it says x=0, 0.1, 0.2, 0.3, ..., 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.Joffan said:Looks good so far.
You could use a table format like this to show the rest of your results:
$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Step }n & y_n & x_n & f(x_n,y_n) & h & \Delta y & y_{n+1} \\
\hline
0 & 2 & 0 & 1 & 0.1 & 0.1 & 2.1 \\
\hline
1 & 2.1 & 0.1 & 4.1428 & 0.1 & 0.41428 & 2.51428 \\
\hline
2 & 2.51428 & 0.2 & ?? & 0.1 & ? & ?? \\
\hline
3 & ?? & 0.3 & ?? & 0.1 & ? & ?? \\
\hline
4 & ?? & 0.4 & ?? & 0.1 & ? & ?? \\
\hline
5 & ?? & 0.5 & ?? & 0.1 & ? & ?? \\
\hline
6 & ?? & 0.6 & ?? & 0.1 & ? & ?? \\
\hline
7 & ?? & 0.7 & ?? & 0.1 & ? & ?? \\
\hline
8 & ?? & 0.8 & ?? & 0.1 & ? & ?? \\
\hline
9 & ?? & 0.9 & ?? & 0.1 & ? & ?? \\
\hline
\end{array}
$$
| Step (n) | yn | xn |
|---|---|---|
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)Math10 said:So the answer starts with y1 to y10, right?
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.
[TD1]0[/TD1][TD1]2[/TD1] [TD1]0[/TD1] [TD1]1[/TD1][TD1]2.1[/TD1] [TD1]0.1[/TD1]
Step (n) yn xn
I don't which I like better either. It's pretty much six of one, half a dozen of the other.Joffan said:Thanks... not sure which I prefer really, but the compact output is good.
Euler's method is a numerical technique used to approximate the solution to a first-order ordinary differential equation (ODE). It involves breaking down the continuous ODE into smaller, discrete steps and using the derivative at each step to calculate the next point.
To use Euler's method with a step size of h=0.1, you would first need to know the initial condition and the derivative at that point. Then, you would use the following formula to find the approximate value at the next point:
yn+1 = yn + h * f(xn,yn)
where h is the step size, xn and yn are the coordinates of the previous point, and f(xn,yn) is the derivative at that point.
Euler's method is used because it provides a simple and efficient way to approximate the solution to an ODE. It is also versatile and can be applied to a wide range of problems, making it a useful tool in many scientific fields.
The benefits of using Euler's method include its simplicity and versatility, as well as its ability to provide a quick approximation of the solution to an ODE. However, it also has limitations, such as its tendency to accumulate error over multiple steps and its inability to accurately handle complex or nonlinear equations.
The accuracy of the approximate values obtained using Euler's method with h=0.1 depends on the complexity of the ODE and the number of steps taken. Generally, the smaller the step size, the more accurate the approximate values will be. However, even with a small step size, Euler's method may not provide highly accurate results for complex or nonlinear equations.