An earlier post discusses Mathematica code for Euler’s method. Here is some updated code.
A couple preliminary notes:
- I like to clear all variables at the beginning.
- For systems it is fun to superimpose the plot on top of the stream plot of the corresponding vector field
Example 1
For the IVP 
Clear["Global`*"]
f[t_, y_] := t - y
deltat = 0.1;
eulerValues =
RecurrenceTable[{t[k + 1] == t[k] + deltat,
y[k + 1] == y[k] + deltat*f[t[k],y[k]],
t[0] == 0.,
y[0] == 2.},
{t, y}, {k, 0, 10}];
Grid[eulerValues]
eulerPlot=ListLinePlot[eulerValues, PlotMarkers -> Automatic]
realPlot = Plot[3 Exp[-t] + t - 1,
{t, 0, 1}, PlotStyle -> Red];
Show[eulerPlot, realPlot]
Example 2
Consider 
We use
Clear["Global`*"]
deltat = .05;
f[y_] := Exp[y];
values = RecurrenceTable[{
t[k + 1] == t[k] + deltat,
y[k + 1] == y[k] + deltat*f[y[k]],
t[0] == 0, y[0] == 1
}, {t, y}, {k, 0, 10}];
Grid[values]
ListLinePlot[values, PlotMarkers -> Automatic]
Example 3
Here we consider the system
with 
Clear["Global`*"]
f[x_, y_] := x (1 - x) - x*y
g[x_, y_] := -y + 2 x*y
streamPlot = StreamPlot[
{f[x, y], g[x, y]},
{x, -1, 2},
{y, -1, 2}]
deltat = 0.1;
steps = 100;
eulerValues = RecurrenceTable[{
t[0] == 0, x[0] == 1.5, y[0] == 1,
t[k + 1] == t[k] + deltat,
x[k + 1] == x[k] + deltat*f[x[k], y[k]],
y[k + 1] == y[k] + deltat*g[x[k], y[k]]},
{t, x, y},
{k, 0, steps}];
eulerPlotValues = Table[{
eulerValues[[k, 2]],
eulerValues[[k, 3]]},
{k, 1, steps + 1}];
eulerPlot = ListLinePlot[
eulerPlotValues,
PlotMarkers -> Automatic,
PlotStyle -> Red]
Show[streamPlot, eulerPlot]
or characteristic functions, one can pretty quickly develop intuition for the idea that begin “concentrated” in one space roughly corresponds to being “spread out” in another.![[0,1]](https://s0.wp.com/latex.php?latex=%5B0%2C1%5D&bg=ffffff&fg=333333&s=0 "[0,1]")
in frequency space. What does it look like in physical space? What happens if we add a function with frequency support in ![[25,30]](https://s0.wp.com/latex.php?latex=%5B25%2C30%5D&bg=ffffff&fg=333333&s=0 "[25,30]")
? Etc.
and are 
, but are not necessarily 
conformally compact. We subsequently investigate the rate at which curvature invariants decay at infinity, identifying a conformally invariant tensor which serves as an obstruction to “higher order decay” of the Riemann curvature operator. Finally, we establish Fredholm results for geometric elliptic operators, extending the work of Rafe Mazzeo and John M. Lee to this setting. As an application, we show that any weakly asymptotically hyperbolic metric is conformally related to a weakly asymptotically hyperbolic metric of constant negative curvature.
conformally compact, but not necessarily 
conformally compact. In order to ensure that the data sets we construct are indeed shear-free, we make use of the conformally covariant traceless Hessian introduced in [arXiv:1506.03399]. We furthermore construct a class of initial data sets with weakly asymptotically hyerbolic metrics that may be only 
conformally compact; these data sets are insufficiently regular to make sense of the shear-free condition.
tends to zero is not because some species isn’t getting enough food, but because the differential equation governing 
is too large” should be avoided. What is more appropriate is something like “if 
for various values of 
. The first thing I would do is plot the function with some sample value, such as 
.
is increasing so long as it is less than 
.
was a solution to the differential equation
independently from the right side 
, and then to compare the two.
and the asymptote at 
. No other points need to be indicated.
satisfying the same differential equation, but with initial value 
.
and stating that both 
and 
are continuous for all values of $y$.
and 
cannot cross. Therefore, since 
, we know that 
so long as the solution exists.
.