**EULER'S **

Ian Bruce

This is the start of a large project that
will take a year or two to complete : yet I feel that someone should do it in
its entirety, since Euler's calculus works are interconnected in so many ways,
as one might expect, and Euler had a habit of returning to earlier ideas and
making improvements. John D. Blanton has already translated Euler's *Introduction to Analysis* and approx. one third of Euler's
monumental *Foundations of Differential
Calculus* : this is not really much help to me, as I would have to refer
readers to texts that might not be available to them, and even initially, I
have had to delve into Ch.18 of Part II of the latter book to obtain
explanations of the formulae used in the first chapter of the integration,
which was our main concern here initially. I have decided to start with the
integration, as it shows the uses of calculus, and above all it is very
interesting and probably quite unlike any calculus text you will have read
already. Euler's abilities seemed to know no end, and in these texts well
ordered formulas march from page to page according to some grand design. I hope
that people will come with me on this great journey : along the way, if you are
unhappy with something which you think I have got wrong, please let me know and
I will fix the problem a.s.a.p. There are of course, things that Euler got
wrong, such as the convergence or not of infinite series; these are put in
place as Euler left them, perhaps with a note of the difficulty. The other
works mentioned are to follow in a piecemeal manner alongside the integration
volumes, at least initially on this web page. These works are available in the
public domain on the Euler Archive website and from Google Books ; I have made the corrections suggested from
time to time by the editors in the *Opera
Omnia* edition, to all of whom I express my thanks. The work is divided as
in the first edition and in the Opera Omnia into 3 volumes. All the chapters
presented here are in the books of Euler's original treatise, which corresponds
to Series I volumes 11,12 & 13 of the *O.O.*
edition. I have done away with the sections and parts of sections as an
irrelevance, and just call these as shown below, which keeps my computer much
happier when listing files.

**The
resolution of differential equations of the second order only.**

Click here for
the 1^{st} chapter **: ***Concerning the integration of simple
differential formulas of the second order. *Use is made of the variables*, p = dy/dx *and* q = dp/dx *in
solving some more difficult first order differential equations. At this stage
the exclusive use of the constant differential *dx*, which can be seen in the earlier work of Euler via Newton is
abandoned, so that *ddx*
need not be zero, and there are now four variables available in solving second
order equations : *p, q, x*, and *y*. Euler admits that this is a more
powerful method than the separation of variables in finding solutions to such
equations, where some differential quantity is kept constant.

Click here for
the 2^{nd} chapter **: ***Concerning second order differential
equations in which one of the variables is absent. *

Further use is made of
the variables*, p = dy/dx
*and* q = dp/dx
*in solving some second order differential equations. The idea of solving
such equations in a step–like manner is introduced; most of the equations
tackled have some other significance, such as relating to the radius of
curvature of some curve, etc.

Click here for
the 3^{rd} chapter **: ***Concerning homogeneous second order
differential equations, and those which can be reduced to that form. *Further
use is made of the variables*, p = dy/dx, q = dp/dx, p = ux, and q = v/x *in solving some homogeneous second
order differential equations. In these examples a finite equation is obtained
between some of the variables, as *x*
disappears. Euler displays his
brilliance in finding integrating factors for these equations, to one of which
I have added a note (§807 Scholium) ; others I have left to the intrepid
investigators of this work.

Click here for
the 4^{th} chapter **: ***Concerning second order differential
equations in which the other variable y has a single dimension. *A careful
exposition is made of equations of the form *y''
+Py' + Qy = X*, where *P, Q*,
and *X* are functions of *x*, written of course in the Euler manner
*ddy** + Pdy* +
...etc. A lot of familiar material is uncovered here, perhaps in an unusual
manner : for example, we see the origin of the particular integral and
complementary function for integrals of this kind.

Click here for
the 5^{th} chapter **: ***Concerning the integration by factors of
second order differential equations in which the other variable y has a single
dimension. *Now equations of the form *y''
+Py' + Qy = X*, where *P, Q*,
and *X* are functions of *x*, are considered that can be solved
completely. The use of multipliers is used in conjunction with the formation of
total differentials, applied in
succession solving such equations for particular forms of *P *and *Q*.

Click here for
the 6^{th} chapter **: ***Concerning the integration of other second
order differential equations by putting in place suitable multipliers. *This
is a harder chapter to master, and more has been written by way of notes by me,
though some parts have been left for you to discover for yourself. The methods
used are clear enough, but one wonders at the insights and originality of parts
of the work. The use of more complicated integrating factors is considered in
depth for various kinds of second order differential equations. How much of
this material is available or even hinted at in current texts I would not know;
it seems to be heading towards integral transforms, where the integral of the
transformed equation can be evaluated, and then the inverse transform effected
: but this latter operation is not attempted here.

Click here for
the 7^{th} chapter **: ***On the resolution of the second order
differential equation ddy +a x^n
ydx^2 = 0 by infinite series. *This chapter is rather labour intensive as
regards the number of formulas to be typed out; however, modern computing makes
even this task easier. The relatively easy task of setting up an infinite
series for the integral chosen is accomplished; after which considerable
attention is paid to series that end abruptly due to the introduction of a zero
term in the iteration, thus providing algebraic solutions. Euler had evidently
spent a great deal of time investigating such series solutions of integrals,
and again one wonders at his remarkable industry. Recall that this book was
meant as a teaching manual for integration, and this task it performed
admirably, though no thought was given to convergence, a charge often laid.

Click here for
the 8^{th} chapter **: ***Concerning the resolution of other second
order differential equation by infinite series. *This chapter is also rather
labour intensive as regards the number of formulas to be typed out; here a more
general second order differential equation is set up and integrated by a series
expansion. The emphasis is now on degenerate cases, which arise when the roots
of the indicial equation are equal or imaginary, and the *ln* function is introduced as a
multiplier of one of the series; there is a desire to obtain the complete
integral for these more trying cases.

Click here for
the 9^{th} chapter **: ***Concerning the resolution of other second
order differential equation of the form*

* Lddy +
Mdxdy +Nydx ^{2}=0.* This is a most
interesting chapter, in which other second order equations are transformed in
various ways into other like equations that may or may not be integrable. It
builds on the previous chapter to some extent, and ends with some remarks on
double integrals, or the solving of such differential equations essentially by
double integrals, a process which was evidently still under development at this
time.

Click here
for the 10^{th} chapter **:
***On the construction of second order
differential equations from the quadrature of curves. *In this chapter there
is a move into functions of two variables. The idea is to take an integral of
some function *V*, treat it as a
function of two variables *x* and *u*, and to form a differential equation
of the form * Lddy + Mdydu +Nydu ^{2}=0 *from this integral by
differentiating within the integral

Click here
for the 11^{th} chapter **:
***On the construction of second order
differential equations sought from the resolution of these by infinite series. *This
chapter follows on from the previous one : more degrees of freedom are
introduced by introducing a series with two–fold coefficients, enabling a more
general differential equation to be tackled, that has been met before. An
integral is established finally for the differential equation, the bounds of
which both give zero for the dummy variable, an artifice that enables
integration by parts to be carried out without the introduction of extra terms.
The variable *x* in the original d.e. is treated as a constant in the integration.

Click here
for the 12^{th} chapter **:
***Concerning the integration of second order differential equations by
approximations. *This chapter is not about what you might think from the
title, and does not offer much in the way of
the approximate evaluation of integrals numerically, even if they are of
second order, apart from advocating the
use of very small intervals, and eventually a more involved way that allows the
second derivative to change in the initial interval is set out for use, and
giving rise to quadratic quadrature over each interval. If anything, the
chapter sets the stage for an iterative program of some kind, and thus is of a
general nature, while what to do in case of diverging quantities is given the
most thought. This marks the end of
Section I.

**The resolution of differential equations
of the third or higher orders which involve only two variables. **

Click here for
the 1^{st} chapter **: ***Concerning the integration of simple
differential formulas of the third or higher orders. *

Euler derives some very pretty results for the integration of these simple higher order derivatives, but as he points out, the selection is limited to only a few choice kinds. Thus the chapter is rather short.

Click here for
the 2^{nd} chapter **: ***Concerning the integration of differential
formulas of this form Ay +By' +Cy'' .... +Ny ^{(n)}
on considering dx constant. *This is the most beautiful of chapters in this
book to date, and one which must have given Euler a great deal of joy ; there
is only one thing I suggest you do, and that is to read it.

Click here for
the 3^{rd} chapter **: ***Concerning the integration of differential formulas
of this form Ay +By' +Cy'' .... +Ny ^{(n)} =
X, on considering dx constant. *This is clearly a continuation of the
previous chapter, where the method is applied to solving

Click here for
the 4^{th} chapter **: ***The application of the method of integration
treated in the last chapter to examples*. The examples are restricted to
forms of *X* above for which the
algebraic equation has well-known roots. Much light is shed on the methods
promulgated in the previous chapter, and this chapter should be read in
conjunction with the preceding two chapters. Euler takes the occasion to extend
*X* to infinity in a Taylor expansion
at some stages.

Click here for
the 5^{th} chapter **: ***Concerning the integration of differential
formulas of this form X=Ay +Bxy' +Cx ^{2}y'' +
etc. *This is a chapter devoted to the solution of one kind of differential
equation, where the integrating factor is simply

Ian
Bruce. June 29^{th},
2010 latest revision. Copyright : I reserve the right to publish this
translated work in book form. However, if you are a student, teacher, or just
someone with an interest, you can copy part or all of the work for legitimate
personal or educational uses. Please feel free to contact me if you wish by
clicking on my name, especially if you have any relevant comments or concerns.