Leonhard Euler: the Beethoven of Mathematics
Andrea Hall
April 19, 2011
Ms. Brewer
Modern Differential Equations
Hall 1
Biography of Leonhard Euler:
Leonhard Euler (1707-83) was a prolific mathematician who is renowned for his
profound influence in many facets of mathematics. Euler had a gift for mathematics and although
his father originally wanted him to pursue theology, Johann Bernoulli convinced him that
mathematics was the right fit for his son. Euler moved from his birthplace of Basel to St.
Petersburg in1725. During his time in St. Petersburg, Euler worked with Daniel Bernoulli a
physicist who fostered Euler’s interest in hydrodynamics and Christian Goldbach who
encouraged his interest in number theory. Euler solved the Basel problem that was previously
only defined by bounds and found the sum to be
𝜋2
6
. Euler became noted for his Euler’s
approximations. These include the Euler-Maclaurin formula and Euler’s method. Euler’s
method will be discussed in more detail later in this paper.
In 1741, Euler decided to become a part of the Prussian Academy of Sciences and moved
to Berlin which marks some of his most productive years. During this period 275 of his works
were published. He also became active in other pursuits such as overseeing the library, botanical
garden, and observatory. Euler’s relation with King Frederick II of Prussia disintegrated and this
caused Euler to return to Russia in 1766. Euler spent his final years in St. Petersburg.
Euler was the founder of mathematical analysis as we know it today. He delved into the
study of functions, power series, and numerical methods. His analysis contributed to practical
uses in sciences and engineering. Euler defined the relation between the exponential function and
trigonometric identities to be 𝑒 𝑖𝜃 = 𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃. The result of Euler’s influence on analysis
was the assimilation of his mathematical notation. Euler’s notations that he defined that are still
used today include f(x) for a function, the trigonometric functions sin(x), cos(x), i to signify the
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√−1, the symbol 𝜋 for pi, the symbol ∑
for summation,
to signify definite differences, and
small lowercase letters for sides of a triangle and capital letters for their relative angles. Euler’s
notation for the base of a natural logarithm, e, also became standard.
One of Euler’s most impressive skills was his ability to explain difficult subject matter in
a manner that people could understand. He redefined the hierarchy for calculus. Euler’s works
explained difficult concepts so that they were no longer reserved for work by only a few
specialists. Euler was able to illustrate his path to a solution and demonstrate the long,
roundabout, and sometimes clumsy process but bring it together in the end to formulate a simple,
graceful solution. Euler’s ability to effectively explain concepts, even basic ones, is manifested
in the fact that he produced texts for Russian schools on elementary mathematics.
Although Euler was primarily a mathematician, he also influenced areas of physics,
mechanics, astronomy, and ballistics to name some. Celestial mechanics was a subject of study
in his later years. He collaborated with the astronomer, Clairaut, to define lunar motion. Euler
had an appetite for engineering topics as well such as hydraulics, ship design, and navigation.
Leonhard Euler was an unpretentious man. He always gave more credit than was
necessary for previous works by other mathematicians and downplayed his contributions. For
example, he taught the mathematician Joseph-Louis Lagrange and waited for Lagrange’s work
on the calculus of variations to become recognized before publishing his own earlier findings on
the same topic.
Euler had prodigious mental capacity. In mathematical as well as other areas his memory
was beyond the average realm. He memorized all of Virgil’s Aeneid and was able to recite the
first and last line of every page. He remembered exact Academy of Sciences minutes years later.
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In the mathematical sense Euler was able to compute complex mathematical series, e.g. sum
series to the 17th term, in his head. He actually enjoyed calculation and for his research often
performed more computation than might have been necessary.
Three years after suffering a near fatal fever in 1735, Euler went blind in his right eye. In
1771, he developed a cataract in his left eye and became for the most part blind. The blind
mathematician carried out complex lunar mechanics calculations in his head in the final years of
his life. It is astonishing to think that Euler was still able to continue his lunar theory research
and plow forward in his exploration without sight and having to rely on his extraordinary mind to
organize and think through difficult concepts as well as execute complex calculations. Just as
Beethoven composed symphonies in his head without hearing them, Euler composed his own
mathematical series in his head without seeing them. In 1783, the “Beethoven of Mathematics”
died painlessly while having a cup of tea.
Hall 4
Explanation of Euler’s Method:
𝐺𝑖𝑣𝑒𝑛
ℎ=
𝑑𝑦
= 𝑓(𝑥, 𝑦)𝑜𝑛 (𝑥0 , 𝑥𝑛 ) 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑢𝑏𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑠:
𝑑𝑥
(𝑥𝑛 − 𝑥0 )
𝐴𝑛 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 𝑡𝑜 𝑦(𝑥𝑛 )𝑐𝑎𝑛 𝑏𝑒 𝑓𝑜𝑢𝑛𝑑 𝑏𝑦: 𝑦𝑛+1 = 𝑦𝑛 + ℎ ∗ 𝑓(𝑥𝑛 , 𝑦𝑛 )
𝑛
Euler’s method is an approximation of the tangent line at a certain point. As illustrated by the
above adjacent graphs, it can be seen that the approximation values are more accurate when the
point is closer to the original 𝑦0 . It can also be noted that with more subintervals closer together,
error can be decreased.
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Example:
Given: y’=-2xy
y(0)=2
on [0, .5]
to approximate y(.5)
n
𝑥𝑛
𝑓(𝑥𝑛 , 𝑦𝑛 )
𝑦𝑛+1
Error=|𝑓(𝑥𝑛 , 𝑦𝑛 ) − 𝑦𝑛 |
0
0
0
2 = 𝑦1
0.019900335
1
0.1
-0.4
1.96 = 𝑦2
0.0384211217
2
0.2
-0.784
1.8616 = 𝑦3
0.0337376295
3
0.3
-1.12896
1.768704 = 𝑦4
0.0644164221
4
0.4
-1.4149632
5
0.5
1.62720768 = 𝑦5 0.0696061139
This table shows a progression of the approximation of the y values with a subinterval of 0.1.
The error is calculated (sample calculation shown below) by first finding the exact equation by
integrating the y’ since this can be integrated by hand in this case (shown below) and subtracting
the approximated value from the real value. It can be seen that error increases as n increases. The
farther away from 𝑦0 , the greater the error becomes. Still, the 𝑦5 approximation value is only
about 0.0696 away from the true value. This is a fairly accurate approximation.
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Sample Calculations:
𝑦1 = 𝑦0 + ℎ(𝑓(𝑥0 , 𝑦0 )
= 2 + (0.1)(10)
=2
f(x, y)
f(0.1,2)
=-2(0.1)(0.2)
= -0.4
Exact Solution
y’= -2xy
1
𝑑𝑦 = −2𝑥 𝑑𝑥
𝑦
2
𝑒 ln(𝑦) = 𝑒 −𝑥 𝑒 𝑐
y(x)=A𝑒 −𝑥
2
y(0)=A𝑒 0 =2 A=2
y(x)= 2𝑒 −𝑥
2
Error
2
y(0.2)=2𝑒 −0.2
2
|2𝑒 −0.2 − 𝑦2 |=0.0384211217
Improved Euler’s Method (Heun’s Method) improves upon Euler’s original method and
decreases error. This method uses the trapezoid rule instead of rectangles and becomes more
accurate.
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Real Life Applications of Euler’s Method:
Euler’s method is a numerical solution. It may be unclear at first why one would use numerical
solutions to solve a problem instead of an exact solution. However, sometimes exact solutions
are extremely difficult to come by or simply impossible. Numerical methods are used to solve
problems when an exact solution is not pragmatic. Euler’s method is a convenient way to solve a
differential equation when integrating is difficult or impossible. Computers use approximation
processes similar to Euler’s method to solve differential equations by a series of iterations. Some
examples of what a computer would use numerical methods to solve in a real life situation would
be falling objects, Newton’s law of cooling, beam deflection, virtual work, finite element
analysis, oscillations, population growth, and exponential decay.
Hall 8
Bibliography
Assad, Arjang A. "Leonhard Euler: A Brief Appreciation." Networks 49.3 (2007): 190-98. Wiley
Online Library. Web. 10 Apr. 2011.
Barker, Christopher A. "Numerical Methods--Euler's Method." Delta College Differential
Equations Lab Home Page. 2009. Web. 10 Apr. 2011.
<http://calculuslab.deltacollege.edu/ODE/7-C-1/7-C-1-h.html>.
"Differential Equations and Euler's Method." Web. 10 Apr. 2011.
<http://http://college.cengage.com/mathematics/larson/calculus_analytic/7e/instructors/do
wnloads/apthemes/2258_12.pdf>.
"The Euler-Heun Method." Http://livetoad.org/Courses/Documents/214a/Notes/eulerheun_method.pdf. Web. 10 Apr. 2011.
"Applications of Differential Equations." Free Mathematics Tutorials, Problems and Worksheets
(with Applets). 26 Nov. 2007. Web. 10 Apr. 2011.
http://www.analyzemath.com/calculus/Differential_Equations/applications.html.