136 DIFFERENTIAL EQUArloNsIn Problems 16 to 18, find a particular solution by using Method 4'16. Y" - 4Y' + 3Y : x3eb'1fl. y" - 7Y' + l2Y : eu(x' - 5")'1E. y" + 2y' + y - 2x'e-b l 3eL.In Problems 19 to 24, find a particular solution by any method'lg. l"' - 8Y : l6x'-20.yQ)-l:l-x3.21. y"' - Iy' : x.D. y$) - *-t.8.y"'-Y"+Y'=x+1.A. y"' r 2y" : 7.25. Use the exponential shift rule to find the general solution of each of thefollowing equations:(a) (D - 2)'y : e- [hint: multiPlY bY e(b) (D + l)3y :12"-''(c) (D - 2)'Y : e} sinx.-l and use (10)l;26. Consider the zth order homogeneous equation p(D)y :0'(a)Ifapolynomialq(r)isafactoroftheauxiliarypolynomialp(r)'showthat any solution of the differential equation q(D)y : 0 is also a solutionof P(D)Y : s'(b) If ; is a root of multiplicity /< of the auxiliary equation p(r) : 0' showthat any solution of (D - r'')oy :0 is also a solution of p(D)y : 6'(c) Use the exponential shift rule to show that (D - 'r)uy : 0 has| = (cr * crx * c.x' + "' * cuxk-t)en"as its general solution. Hint: (D - n)-y : O is equivalent toeryDkk-\'y) :0.APPENDIX A. EULERLeonhard Euler (1707-1783) was Switzerland's foremost scientist andone of the three greatest mathematicians of modern times (the other twobeing Gauss and Riemann).He was perhaps the most prolific author of all time in any field.From 1727 to tzsg his writings poured out in a seemingly endless flood,constantly adding knowledge to every known branch of pure and appliedmathemaiics, and also to many that were not known until he createdthem. He averaged about 800 printed pages a year throughout his longlife, and yet he il*ort always had something worthwhile to say and neverseems long-*inded. The publication of his complete works was started in1911, and the end is not in sight. This edition was planned to include 887titles in 72 volumes, but since that time extensive new deposits ofpreviously unknown manuscripts have been unearthed, and it is nowND ORDER LINEAR EQUATIONS 137estimated that more than 1fi) large volumes will be required forcompletion of the project. Euler evidently wrote mathematics with theease and fluency of a skilled speaker distoursing on subjects with whichhe is intimately familiar. His writings irt'irrcdels of relaxed clarity. Henever condensed, and he reveled in the.rich abundance of his ideas andthe vast scope of his interests. The French physicist Arago, in speaking ofEuler's incomparable mathematical facility, remarked that "He calcu-lated without apparent effort, as men breathe, or as eagles sustainthemselves in the wind." He suffered total blindness during the last 17years of his life, but with the aid of his powerful memory and fertileimagination, and with helpers to write his books and scientific papersfrom dictation, he actually increased his already prodigious output ofwork.Euler was a native of Basel and a student of John Bernoulli at theUniversity, but he soon outstripped his teacher. His working life wasspent as a member of the Academies of Science at Berlin and St.Petersburg, and most of his papers were published in the journals ofthese organizations. His business was mathematical research, and heknew his business. He was also a man of broad culture, well versed in theclassical languages and literatures (he knew the Aeneid by heart), manymodern languages, physiology, medicine, botany, geography, and theentire body of physical science as it was known in his time. However, hehad little talent for metaphysics or disputation, and came out second bestin many good-natured verbal encounters with Voltaire at the court ofFrederick the Great. His personal life was as placid and uneventful as ispossible for a man with 13 children.Though he was not himself a teacher, Euler has had a deeperinfluence on the teaching of mathematics than any other man. This cameabout chiefly through his three great treatises: Introductio in AnalysinInfinitorum (1748); Institutiones Calculi Differentialis (1755); andInstitutiones Calculi Integralis (I768-1794). There is considerable truth inthe old saying that all elementary and advanced calculus textbooks since1748 are essentially copies of Euler or copies of copies of Euler.la Theseworks summed up and codified the discoveries of his predecessors, andare full of Euler's own ideas. He extended and perfected plane and solidanalytic geometry, introduced the analytic approach to trigonometry, andwas responsible for the modern treatment of the functions log x and e'.He created a consistent theory of logarithms of negative and imaginarynumbers, and discovered that logx has an infinite number of values. Itrosee C. B. Boyer, "The Foremost Textbook of Modern Times," Am. Math. Monthty,Vol. 58, pp. 223-226, 1951.1-38 DTFFERENTTAL EeuATIoNSwas through his work that the symbols e, n, and i (= VJ) becamecommon currency for all mathematicians, and it was he who linked themtogether in the astonishing relation eni : -I. This is merely a specialcase (put 0 : n) of his famous formula e'u : cos g * i sin 0, whichconnects the exponential and trigonometric functions and is absolutelyindispensable in higher analysis.l5 Among his other contributions tostandard mathematical notation were sin.r, cos.r, the use of /(x) for anunspecified function, and the use of X for summation.16 Good notationsare important, but the ideas behind them are what really count, and inthis respect Euler's fertility was almost beyond belief. He preferredconcrete special problems to the general theories in vogue today, and hisunique insight into the connections between apparently unrelated for-mulas blazed many trails into new areas of mathematics which he left forhis successors to cultivate.He was the first and greatest master of infinite series, infiniteproducts, and continued fractions, and his works are crammed withstriking discoveries in these fields. James Bernoulli (John's older brother)found the sums of several infinite series, but he was not able to find thesum of the reciprocals of the squares, 1 + + + rt + * + "' . He wrote,"If someone should succeed in finding this sum, and will tell me about it,I shall be much obliged to him." ln1736,long afterJames's death, Eulermade the wonderful discovery thatHe also found the sums of the reciprocals of the fourth and sixth powers,r11In2f+-r-+-+'..:-'49t66i11_+., :l-l--+..: I T *t rra11l-241l -r--+'26l1lnor--L. :l+--L-+...:-'34 ' 16'81 90and6Jt'945't An even more astonishing consequence of his formula is the fact that an imaginary