Question

how does one compute the following:

(d^2y/dt^2)-t-y=0
Please use the Huen method
(modified Eulers)

Answer 1

my question is how do I advance the first order derivative?

Answer 2

start with \[w(y,t)={dy\over dt}\\
{dw\over dt}-t-y(t)=0\]
this form can now be implemented using Euler's or modified Euler's or any other linear approximation methods.

Answer 3

I agree thank you.
how to I advance w?

Answer 4

treat the new differential as a separate problem between two iterations of "y"

Answer 5

so, w advances, then y advances, then w again.. and then y again.. and so on

Answer 6

Ah, so yo then wo
then y1 and w1
Thank you

Answer 7

yep

Answer 8

think of "y" as position
then \(w={dy\over dt}\)is your velocity
and \({dw\over dt}={d^2y\over dt^2}\)is your acceleration function.

Answer 9

notice that y -> function of "t"
and w-> function of "t" and "y"

Answer 10

@garrett_payne now, how about being courteous and award?

Answer 11

lol, just joined.
How do I do that?

Answer 12

jsut say best response :)

Answer 13

issue; what happens when dy/dt is 0
the problem is as follows
\[\frac{ d ^{2} y}{ dt ^{2} }-t+y = 0 \]
I haved called \[\frac{dy }{ dt }=z \]
and step size is 0.1
y(0)=2
z(0)=0

Answer 14

ok

Answer 15

could you show 1 iteration?

Answer 16

ok
\[
h=\text{step size}\\
\frac{dz}{dt}=t-y=f_1(t,y)\\
\frac{dy}{dt}=z(t,y)
\]
k^th Iteration steps:
\[
t_{k+1}=k_k+h\\
y_{k+1}=y_k+hz_k\qquad\qquad\text{(first approx.}\\
z_{k+1}=z_k+\frac{h}{2}\left[f_1(t_k,y_k)+f_1(t_{k+1},y_k)\right]\\
y_{k+1}=y_k+\frac{h}{2}\left[z_k+z_{k+1}\right]\qquad\qquad\text{(second approx.}
\]

Answer 17

correction... in the second term of \(z_{k+1}\), I wrote \(y_k\) by mistake instead of \(y_{k+1}\)

Answer 18

I caught that, Thank you so much
My numerical operations prof is an retrice

Answer 19

for accuracy, you may choose to perform the last two steps twice to improve accuracy

Answer 20

do you mean, replugging the obtained value in the same equation?

Answer 21

yes. coz, we approximated the y_{k+1} the first time...