euler.nb
1
Euler Method Mathematica Lab
After a brief introduction and warm-up exercise on the basics of Mathematica we create a
small routine for approximating the solution to a first order ordinary differential equation
using the Euler method.
The first thing you will probably notice about Mathematica is that it is finickey: it is case sensitive and things must be typed with proper syntax. Aside from being a symbolic math package,
Mathematica is a full programming language.
The workspace is divided into cells. All commands in a cell are evaluated by typing SHIFT+RETURN. It is generally best to use lower case letters for variables since many upper case
letters are reserved for operators. Operators start with capital letters and expressions being
operated on go inside square brackets: Sin[x] or Factor[x^2-4]. Use a space for multiplication
so f r e d is f times r times e times d whereas fred is recognized as a single variable. The
help browser is extremely useful. Click on the small triangle at the bottome to see the further
examples. I usually look there first for a problem similiar to the one I'm trying to solve. I generally copy and paste the example into my worksheet and modify it for my problem. Here are
some examples of what you can do with Mathematica:
Examples of some basic tasks using Mathematica.
In the following cells I multiply 4 and 17, factor x^2-6x+9, compute the derivative of (x^2+Cosh[x])/Exp[Sin[x]], and find the integral of Exp[a x] Sin[b x] (I got a complex answer so I
use the Simplify command to get a simpler answer). Try them out yourself!
4 17
68
Factor!x ^ 2 ! 6"x # 9"
!!3 " x"2
D!#x ^ 2 # Cosh!x"$ % Exp!Sin!x"", x"
!#!Sin#x$ Cos#x$ !x2 " Cosh#x$" " #!Sin#x$ !2 x " Sinh#x$"
Integrate!Exp!a x" Sin!b x", x"
a #a x Sin#b x$
b #a x Cos#b x$
! $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$$$$$$$$$$$ " $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$$$$$$$$$$$
!a ! % b" !a " % b"
!a ! % b" !a " % b"
euler.nb
2
Simplify!Integrate!Exp!a x" Sin!b x", x""
!a x !"b Cos"b x# # a Sin"b x#$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$$$$$$$$$$$
a2 # b2
The command Clear[x,y,t,f] clears all previous assignments to the variables x, y, t, and f.
Example:
x ! fred
fred
x
fred
Clear!x"
x
x
The strange notation /. x!3 is used to evaluate an expression at 3. Example:
Sin!x" Exp!"3 x" #. x # "2 Pi # 3
1 %&&&
" $$$$ 3 !2 %
2
Using Mathematica to solve algebraic and differential equations analytically.
Mathematica can be used to find solutions to equations. Algebraic equations are solved using
Solve and differential equations are solved using DSolve. Notice double equal signs are
used for conditional equalities. Examples:
Solve!x ^ 2 $ 3%x $ 3 & 0, x"
1
1
%&&&
%&&&
''x & $$$$ ("3 " ' 3 )*, 'x & $$$$ ("3 # ' 3 )**
2
2
Here's the equation y' = - y. We get the general solution with arbitrary constant C[1]. It is important to clear all variables before using DSolve.
Clear!y, t"
DSolve!y '!t" & "y!t", y!t", t"
++y"t# & !"t C"1#,,
Here is an initial value problem y' = - y, y(0)=2:
euler.nb
3
Clear!y, t"
DSolve!#y '!t" ! "y!t", y!0" ## 2$, y!t", t"
!!y"t# ! 2 "#t $$
We can use Plot to make a graph of a solution curve; I copied and pasted the output from the
previous example into the slot for the expression:
Plot!2 $"t , #t, 0, 5$"
2
1.5
1
0.5
1
2
3
4
5
Implementing Euler's method for solving an ODE numerically.
We can build routines using loops to carry out iterative procedures such as implementing the
Cauchy-Euler algorithm. Look in the help browser for Programming/Flow Control for instructions on using the commands Do, While, etc. In the following example I made a do loop to
approximate the value of y[1] four the initial value problem y' = 1- t + 4 y, y[0]=1. I took h =
0.01 so there are (1-0)/0.01 = 100 iterates in this example. In other words, the y-value we seek
is y[99+1]. Notice that in this example the initial value is given at t=0 so t = 0+0.01 n = 0.01
n in our algorithm.
In[119]:= Clear!y"
y!0" # 1;
Do!y!n % 1" # y!n" % .01 %1 " .01 n % 4 y!n"&, #n, 0, 99$"
y!100"
Out[122]= 60.0371
The approximate value of y[1], using h=.01, is 60.0371. We can make a table of the approximate solution values (t,y(t)) and use them to plot the approximate solution curve using ListPlot
to graph a curve through these points. (I named the graph "a" so I can use it again later.) The
first line makes the table and gives it the clever name 'table'. The second line plots the points
and joins them with a curve. If we need some of the values (for the homework, for instance)
we can read them off of the table.
euler.nb
4
In[123]:= table ! Table!"N!.01 k#, y!k#$, "k, 0, 100$#
a ! ListPlot!table, PlotJoined " True#
Out[123]= !!0., 1", !0.01, 1.05", !0.02, 1.1019", !0.03, 1.15578", !0.04, 1.21171", !0.05, 1.26978",
!0.06,
!0.11,
!0.16,
!0.21,
!0.26,
!0.31,
!0.36,
!0.41,
!0.46,
!0.51,
!0.56,
!0.61,
!0.66,
!0.71,
!0.76,
!0.81,
!0.86,
!0.91,
!0.96,
1.33007", !0.07, 1.39267", !0.08, 1.45768", !0.09, 1.52518", !0.1, 1.59529",
1.6681", !0.12, 1.74373", !0.13, 1.82227", !0.14, 1.90387", !0.15, 1.98862",
2.07667", !0.17, 2.16813", !0.18, 2.26316", !0.19, 2.36188", !0.2, 2.46446",
2.57104", !0.22, 2.68178", !0.23, 2.79685", !0.24, 2.91642", !0.25, 3.04068",
3.16981", !0.27, 3.304", !0.28, 3.44346", !0.29, 3.5884", !0.3, 3.73903",
3.8956", !0.32, 4.05832", !0.33, 4.22745", !0.34, 4.40325", !0.35, 4.58598",
4.77592", !0.37, 4.97336", !0.38, 5.17859", !0.39, 5.39193", !0.4, 5.61371",
5.84426", !0.42, 6.08393", !0.43, 6.33309", !0.44, 6.59211", !0.45, 6.8614",
7.14135", !0.47, 7.43241", !0.48, 7.735", !0.49, 8.0496", !0.5, 8.37669",
8.71675", !0.52, 9.07032", !0.53, 9.43794", !0.54, 9.82015", !0.55, 10.2176",
10.6308", !0.57, 11.0604", !0.58, 11.5071", !0.59, 11.9716", !0.6, 12.4546",
12.9567", !0.62, 13.4789", !0.63, 14.0219", !0.64, 14.5864", !0.65, 15.1735",
15.7839", !0.67, 16.4187", !0.68, 17.0787", !0.69, 17.7651", !0.7, 18.4788",
19.2209", !0.72, 19.9927", !0.73, 20.7952", !0.74, 21.6297", !0.75, 22.4975",
23.3999", !0.77, 24.3383", !0.78, 25.3141", !0.79, 26.3289", !0.8, 27.3841",
28.4815", !0.82, 29.6227", !0.83, 30.8094", !0.84, 32.0434", !0.85, 33.3268",
34.6614", !0.87, 36.0492", !0.88, 37.4925", !0.89, 38.9934", !0.9, 40.5542",
42.1774", !0.92, 43.8654", !0.93, 45.6208", !0.94, 47.4463", !0.95, 49.3448",
51.3191", !0.97, 53.3722", !0.98, 55.5074", !0.99, 57.7279", !1., 60.0371""
60
50
40
30
20
10
0.2
0.4
0.6
0.8
1
Out[124]= !"Graphics"!
How did we do?
We can plot the actual integral curve and the approximate integral curve on the same set of
axes. I used DSolve to solve the equation then pasted the result into the Plot command. I called
the approximate plot 'a' and the actual plot 'b'. The command Show combines the graphics.
Clear!y, x#
DSolve!"y '!x# !! 4 y!x# # 1 $ x, y!0# !! 1$, y!x#, x#
1
##y$x% # $$$$$$$ &%3 & 19 '4 x & 4 x'((
16
euler.nb
65
b ! Plot! """"""" "#3 $ 19 %4 x $ 4 x#, $x, 0, 1%&
16
y!0" ! 0;
Do!y!n " 1" ! y!n" " .05 #3 #1 " .05 n$ ^ 2 % #3 y!n" ^ 2 # 4$$, &n, 0, 15'"
60 y!16"
50 table ! Table!&N!1 " .05 k", y!k"', &k, 0, 16'"
ListPlot!table, PlotJoined $ True"
1 table"
In[131]:= Clear!y,
40
Out[134]= !3.0981
30
Out[135]= !!1., 0", !1.05, !0.0375", !1.1, !0.0788874", !1.15, !0.124475", !1.2, !0.174652",
20 !1.25, !0.229916", !1.3, !0.290929", !1.35, !0.3586", !1.4, !0.434238",
10
!1.45, !0.519845", !1.5, !0.618731", !1.55, !0.737089", !1.6, !0.88914",
!1.65, !1.12497", !1.7, !3.13341", !1.75, !3.11638", !1.8, !3.0981""
0.2
Show'a, b(
-0.5
0.4
1.2
0.6
0.8
1.6
0.6
0.8
1.4
1
1.8
60-1
-1.5
50
40-2
-2.5
30
20-3
10
Out[136]= "#Graphics#"
0.2
0.4
1
Your turn! Modify the above code to solve the following problems:
Another example.
As another example let's solve y'=3t^2/(3y^2-4), y(1)=0 for y(1.8) using h=0.05. Here n will
Consider
the initial
value
problem
1.1.Apply
Euler's
method
to solve
y'[t] = y[t] (2-y[t]), y[0]=0.1 for y[4]. Use h=0.2. Make a
go from 0 to (1.8-1)/0.05 = 16. Here our initial value is at t=1 so t=1+0.05 n. Notice, we are
y’[t]curve.
= 1-y[t],
y[0]What
= 2. should the range of n be to get the
table and plot the approximate solution
(Hint:
computing
y[n]'s
not
y[t]'s. Inofthis
example
y[16]thecorresponds
to the y-value at t=1.8.
a)
Use
separation
variables
to
find
solution.
solution at t=4 starting at t=0?)
b)  Apply Euler’s method (in Mathematica) to plot the approximate solution for t = 0
to t = 4. Use a step size of 0.1 Plot both the solution and approximate solution on
2. y'[t] = -0.5 the
y-t+1.5,
y(0)=1
for y(3). Use h=0.01. Make a table (put a semicolon after the
same set
of axes.
table linec) to hide
it's actual
big!) and
plot
theWhat
approximate
solution
curve.
Solve
the using
equation
Whatit is- the
value
y(4)?
is approximate
value
of y(4)
(from
using DSolve.Euler’s
Plot both
the solution
and What
approximate
solution
on the same set of axes.
method
in part b))?
is the relative
error?
2. Consider the initial value problem
y’[t] = cos(t)-y[t], y[0] = 5.
a)  Apply Euler’s method (in Mathematica) to plot the approximate solution for y[8].
Start with a step size of 1, then use successively smaller step sizes. Report what
happens in these trials. Discuss how close your approximate solution is to the
Solutions: actual solution and how the error changes as the step size decreases.
Problem 1.