Solving Eulers Method: Step-By-Step Guide

[Kayne]

Hi All,

Just wanted to check to see if I am doing this correctly

If the equation is

$$F(t) = m\ddot{y}+c\dot{y}+ky$$
$$\dot{y}(0) = 0$$

Where m = 2, c =180, k=25, F(t) = -500sin(10t)

I know that it has to be in the form of

$$yk+1 = yk+f(xk,yk)h$$

So solving Eulers method to 0.2 with a step size of 0.1


$$-500sin(10x)= 2\ddot{y}+180\dot{y}+25y$$

Now to change into for f(x,y) I have done the following

$$f(x,y) = -500sin(10x)-2\ddot{y}-180\dot{y}-25y$$

Is this the correct equation for f(x,y)??

So for i = 0

$$yk+1 = yk+f(xk,yk)h$$
$$y1 = yo+f(xo,yo)h$$

where
x0 = 0
y0 = 0
h = 0.1

This is where I think I have made a mistake becuase the two answers that I have found are very different from one another. I would like to know if I have done this correclty.

$$y1 = 0 + f(0, 0)0.1$$
$$y1 = 0 + f(-500sin(10*0)-2*0-180*0-25 )*0.1$$
$$y1 = 2.5$$

and for i = 1

$$y2 = y1+f(x1,y1)h$$
$$y1 = 1 + f(1, 1)0.2$$
$$y1 = 1 + f(-500sin(10*1)-2*1-180*1-25 )*0.2$$
$$y1 = -58.76$$

Have I used eulers method correctly to solve for y1 = -2.5, y2 = -58.76

Thanks for your time

 

[gtoguy97]

and help!! </code>No, you have not used Euler's method correctly. Euler's method is used to solve a differential equation of the form y' = f(x,y). In this case, your equation is F(t) = m\ddot{y}+c\dot{y}+ky, which is a second order differential equation. You cannot use Euler's method to solve this directly.You will need to use a numerical method such as Runge-Kutta or Adams-Bashforth to solve this equation. These methods are more accurate than Euler's method and can handle higher order equations. For more information on these methods, you should consult a textbook on numerical analysis. Hope this helps!