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How to solve a non linear ODE by Finite Difference Method?
I’m trying to solve the ODE $ y” + y = e^{x} $ by the finite difference method. The details of the procedure are as follow:
$$ y_{k+1} – y_k = h f_k $$
\begin{align*}
f_0 &= 0\\
f_k &= \frac{e^{x_{k+1}} – e^{x_k}}{h}
\end{align*}
The finite difference can be then written as follows:
$$ y_k + \left( y_{k+1} – y_k \right) = y_{k+1} – f_k $$
Therefore, \begin{align*}
y_{k+1} – y_k &= e^{x_k} – h\frac{e^{x_k}-e^{x_{k+1}}}{h}\\
&= e^{x_k}( 1 – h\frac{1-e^{x_{k+1}}}{h} )\\
&= \frac{e^{x_k}-e^{x_{k+1}}}{1 – e^{x_{k+1}}}
\end{align*}
Now, it seems that the $\frac{1-e^{x_{k+1}}}{h}$ is a polynomial of $e^{x_k}$, and therefore I can’t go further. Any suggestions?
A:
I think you can apply the Taylor’s Theorem for functions that converge to a root.
Applying the form of the theorem to $f(x) = e^x$ in the interval $(0,1)$ gives $$f(x) = f(0
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It’s the end of an era for a neighborhood icon. A newspaper box is being relocated in the Bypass, and some residents aren’t happy about it.
KQOW’s Taylor Delahaye has the details.
The owners of the Bypass told residents the newspaper box on Elgin Street is being relocated because it’s too close to their building.
“It’s going to be moved to the middle of the street, and that’s a dangerous thing,” said Wayne Kraus, who lives in the Bypass.
“I think they should bring them back. It’s a great part of the neighborhood. The neighbors enjoy it. It brings life to the street, and I think it’s important.”
Kraus, like most other neighbors, says the change will affect drivers and pedestrians, especially at night.
One of those drivers says she’s not happy about the change.
“I don’t like it. I feel like it’s an inconvenience for me, yes. I think it’s a nice part of the neighborhood,” said Alana Pace. “So many people shop at the little box. You know what I mean?”
Pace says she thinks the owners might move into the Bypass because the newspaper box is too far from their building.
“It’s going to be further down the street, and I think that’s ridiculous,” Pace said.
“The city needs to put a road building that goes down the whole street right there. All of a sudden, the whole neighborhood would be better. We need a road on that side of the street.”
“That’s going to bring traffic to a screeching halt,” Neighbors say.
“I just think this
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