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

[karush]

$\tiny{2.7.2}$
1000
(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 $

 

[I like Serena]

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}

 

[karush]

ok I am still processing understanding this saw this on another example
not sure what n is

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

 

[I like Serena]

ok I am still processing understanding this saw this on another example
not sure what n is

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

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

 

[karush]

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

 

[karush]

.

 

[topsquark]

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

 

[karush]

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

 

[topsquark]

well this was posted several years ago
I was just testing a link

Hah! I fell for a necro-post! :)

-Dan

 

[karush]

well MHB is my social life right now...
the gestapo is everywhere...