-b.2.7.2 Euler's method y'=3+y-y y(0) =1

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    Euler's method Method
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Discussion Overview

The discussion revolves around the application of Euler's method to solve an initial value problem represented by the differential equation \(y' = 3 + t - y\) with the initial condition \(y(0) = 1\). Participants explore various step sizes for the method and compare the results obtained at specific time points.

Discussion Character

  • Mathematical reasoning
  • Homework-related
  • Exploratory

Main Points Raised

  • Post 1 presents the initial value problem and outlines tasks involving Euler's method with different step sizes \(h = 0.1\), \(0.05\), and \(0.025\), along with a request for comparison of results.
  • Post 2 defines Euler's method and provides calculations for the first step using \(h = 0.1\), yielding \(y_1 = 1.2\).
  • Post 3 expresses confusion about the notation used in Euler's method, specifically regarding the variable \(n\).
  • Post 4 reiterates the confusion about the notation and seeks clarification on the function \(f\) that describes \(y'\).
  • Post 5 confirms the explanation about the function \(f\) and its relation to \(y'\), indicating an attempt to understand the method better.
  • Post 7 and Post 8 address a participant's learning process, emphasizing the importance of understanding the problem before attempting to solve it using Euler's method.
  • Post 9 humorously acknowledges a delayed response to an older post, indicating a casual tone in the discussion.
  • Post 10 reflects on the social aspect of the forum, suggesting a personal connection to the community.

Areas of Agreement / Disagreement

The discussion includes varying levels of understanding regarding Euler's method and its application. Some participants express confusion about the notation and the method itself, while others provide clarifications. There is no consensus on the best approach to the problem, and the discussion remains exploratory.

Contextual Notes

Participants express uncertainty about the notation and the steps involved in applying Euler's method, indicating a need for further clarification on these points. The discussion also reflects a mix of familiarity with the topic among participants.

karush
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(a) Find approximate values of the solution of the given initial value problem\\
at $t = 0.1, 0.2, 0.3$, and $0.4$ using the Euler method with $h = 0.1$.(b) Repeat part (a) with h = 0.05. Compare the results with those found in (a).(c) Repeat part (a) with h = 0.025. Compare the results with those found in (a) and (b).(d) Find the solution $y=\phi(t)$ of the given problem and evaluate
$\phi(t)$ at $t = 0.1,\quad 0.2, \quad 0.3,$ and $0.4$.#1. $\quad\displaystyle
y'=3+t-y \quad y(0)=1$ok assume first step is to get a general solution
rewrite
$y'+y=3+t $
then
$ey'+ey=(ey)'=3+t$
so
$\displaystyle ey=\int{(3+t)} \, dt= \frac{t^2}{2} + 3 t + c$
isolate
$\displaystyle y=\frac{t^2}{2e} + 3 te^{-1} + c^{-e}$
$\color{red}{(a) 1.2, 1.39, 1.571, 1.7439}$
$\color{red}{(b) 1.1975, 1.38549, 1.56491, 1.73658}$
$\color{red}{(c) 1.19631, 1.38335, 1.56200, 1.73308}$
$\color{red}{(d) 1.19516, 1.38127, 1.55918, 1.72968}$Red is book answer
If I can get #1 probably 2,3 and 4 will be a slide
which are
2. $\quad y'=2y-1 \quad y(0)=1$
3. $\quad\displaystyle
y'=y'=0.5-t+2y, \quad y(0)=1$
4. $\quad\displaystyle
3\cos{t} -2y \quad y(0)=0 $
 
Last edited:
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Euler's method is:
\begin{cases}
t_{k+1}&=t_k+h \\
y_{k+1}&=y_k+h\cdot y'(t_k; y_k)
\end{cases}
At $t_0=0$ we have $y_0=1$, so with $y'=3+t-y$ and $h=0.1$ we have at $t_1$:
\begin{cases}
t_1=t_0+h=0.1 \\
y_1=y_0 + 0.1\cdot y'(0;1)=1 + 0.1 (3+0-1)=1.2
\end{cases}
 
ok I am still processing understanding this saw this on another example
not sure what n is

$$\displaystyle y_{n+1}=y_n +\textit{$hf_n$}$$:confused:
 
karush said:
ok I am still processing understanding this saw this on another example
not sure what n is

$$\displaystyle y_{n+1}=y_n +\textit{$hf_n$}$$:confused:

Close enough.
The function $f$ describes $y'$. That is, we have:
$$y'(t) = f(t,y)$$
And $f_n$ is its value at $t_n, y_n$.
 
Klaas van Aarsen said:
Close enough.
The function $f$ describes $y'$. That is, we have:
$$y'(t) = f(t,y)$$
And $f_n$ is its value at $t_n, y_n$.

Ok I'll try the next one
See if i don't derail
 
.
 
Last edited:
karush, I know you are still learning new material. I can understand that will cause you some difficulties... it's learning.

But how can you pick out a question to put in your collection without knowing the idea of the problem? Euler's method is an approximation method to solve DEqs. You don't need to actually solve the thing until step d and you said you assumed you needed to start by doing it!

-Dan
 
topsquark said:
karush, I know you are still learning new material. I can understand that will cause you some difficulties... it's learning.

But how can you pick out a question to put in your collection without knowing the idea of the problem? Euler's method is an approximation method to solve DEqs. You don't need to actually solve the thing until step d and you said you assumed you needed to start by doing it!

-Dan
well this was posted several years ago
I was just testing a link
 
karush said:
well this was posted several years ago
I was just testing a link
Hah! I fell for a necro-post! :)

-Dan
 
  • #10
well MHB is my social life right now...
the gestapo is everywhere...
 

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