By Kreyszig E.
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Sample text
14. 2x sin x is not listed in the table because it is of minor practical importance. However, by looking at its derivatives, we see that yp ϭ Kx cos x ϩ Mx sin x ϩ N cos x ϩ P sin x should be general enough. Indeed, by substitution and collecting cosine and sine terms separately we obtain (1) (2) (2K ϩ 2Mx ϩ 2P ϩ 2M) cos x ϭ 0 (Ϫ2Kx ϩ 2M Ϫ 2N Ϫ 2K) sin x ϭ 2x sin x. In (1) we must have 2Mx ϭ 0; hence M ϭ 0 and then P ϭ ϪK. In (2) we must have Ϫ2Kx ϭ 2x; hence K ϭ Ϫ1, so that P ϭ 1 and from (2), finally, Ϫ2N Ϫ 2K ϭ 0, hence N ϭ 1.
In doing so, we can set c ϭ ec* and have yϪa ᎏ ϭ ce(a؊b)kt. yϪb We denote the right side by E and solve algebraically for y; then y Ϫ a ϭ (y Ϫ b)E, y(1 Ϫ E) ϭ a Ϫ bE and from the last expression we finally have a Ϫ bE yϭ ᎏ. 1ϪE 42. Let the tangent of such a curve y(x) at (x, y) intersect the x-axis at M and the y-axis at N, as shown in the figure. Then because of the bisection we have OM ϭ 2x, ON ϭ 2y, where O is the origin. Since the slope of the tangent is the slope yЈ(x) of the curve, by the definition of a tangent, we obtain yЈ ϭ ϪON/OM ϭ Ϫy/x.
We have three cases; this is similar to the situation for constant-coefficient equations, to which the Euler–Cauchy equation can be transformed (Team Project 16); however, this fact is of theoretical rather than of practical interest. Comment on Footnote 4 Euler worked in St. Petersburg 1727–1741 and 1766–1783 and in Berlin 1741–1766. He investigated Euler’s constant (Sec. 6) first in 1734, used Euler’s formula (Secs. 6) beginning in 1740, introduced integrating factors (Sec. 4) in 1764, and studied conformal mappings (Chap.