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 1st 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 2nd 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 3rd 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 4th 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 5th 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 6th 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 7th 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 8th 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 9th chapter : Concerning the resolution of other second order differential equation of the form
Lddy + Mdxdy +Nydx2=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 10th 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 +Nydu2=0 from this integral by differentiating within the integral. This is set equal to a chosen function U, which is itself differentiated w.r.t. x. enabling the coefficients L, M, and N to be determined. Thus, the differential equation becomes equal to a function U with the limits w.r.t. x taken, so that U is a function of u only. Essentially the work proceeds backwards from a solution to the responsible differential equation. These details are sketched here briefly, and you need to read the chapter to find out what is going on in a more coherent manner. A number of examples of the procedure are put in place, and the work was clearly one of Euler's ongoing projects.
Click here for the 11th 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 12th 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 1st 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 2nd 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 3rd 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 y for some function of X, using the exponential function with its associated algebraic equation. Serious difficulties arise when the algebraic equation has multiple roots, and the method of partial integration is used; however, Euler tries to get round this difficulty with an arithmetical theorem, which is not successful, but at least provides a foundation for the case of unequal roots, and the subsequent work of Cauchy on complex integration is required to solve this difficulty. This is a long chapter, and I have labored over the translation for a week; it is not an easy document to translate or read; but I think that it has been well worth the effort.
Click here for the 4th 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 5th chapter : Concerning the integration of differential formulas of this form X=Ay +Bxy' +Cx2y'' + etc. This is a chapter devoted to the solution of one kind of differential equation, where the integrating factor is simply xdx. Simple solutions are considered initially for distinct real roots, which progress up to order five. Most of the concern as we proceed is about real repeated roots, which have diverging parts that are shown to cancel in pairs, and complex conjugate pairs, which are easier to handle, and the general form of the solution is gradually evolved by examining these special cases, after which terms are picked out for parts of the general integral. This is the end of Euler's original Book One.
Ian Bruce. June 29th, 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.