How is the solution for y[n+1] obtained using Euler's forward method?

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In summary, the conversation is about solving a differential equation using the Euler (forward) method. The result is obtained by approximating the equation and using a step size and initial value. The solution is obtained through a simple plug-and-chug method.
  • #1
ChickenChakuro
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Hi all, I'm having trouble understanding a basic concept introduced in one of my lectures. It says that:

To solve the DE
[tex]y(t) + \frac{dy(t)}{dt} = 1[/tex] where [tex]y(t) = 0[/tex],

using the Euler (forward) method, we can approximate to:

[tex]y[n+1] = T + (1-T)y[n] [/tex] where [tex]T[/tex] is step size and [tex]y[0] = 0[/tex].

I have no idea how this result is obtained, the only thing they say is that in general for

[tex]\frac{dx_1}{dt} = \frac{x_1[n+1] - x_1[n]}{T}[/tex] for [tex]t = nT[/tex].

Can anyone please help me understand how they arrived at the solution for [tex]y[n+1][/tex]? Thanks!
 
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  • #2
Bah, it is simple plug-and-chug. Should have known! Thanks!
 
  • #3
Four minutes! You didn't even give us a chance to explain!
 

FAQ: How is the solution for y[n+1] obtained using Euler's forward method?

1. What is the Euler forward equation?

The Euler forward equation is a mathematical formula used to approximate the solution of a first-order ordinary differential equation. It is a finite difference method that uses the derivative at the current point to estimate the value at the next point.

2. How is the Euler forward equation derived?

The Euler forward equation is derived from the Taylor series expansion of a function. By truncating the series at the first term, the equation can be simplified to an approximation of the original function at the next point.

3. What are the limitations of the Euler forward equation?

The Euler forward equation is a first-order method and can only approximate the solution of a differential equation accurately if the step size is small. It also tends to accumulate error over time, leading to less accurate results. Additionally, it is only applicable to problems with a single independent variable.

4. What are the applications of the Euler forward equation?

The Euler forward equation is commonly used in physics, engineering, and other scientific fields to solve initial value problems. It is also an essential component of more complex numerical methods, such as the Runge-Kutta method, that are used to solve differential equations.

5. How does the Euler forward equation compare to other numerical methods?

The Euler forward equation is a simple and straightforward method for approximating the solution of a differential equation. However, it is less accurate than more advanced methods, such as the Runge-Kutta method, which can provide more precise results. The choice of method depends on the complexity of the problem and the desired level of accuracy.

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