Differential Equations and Mathematical Physics by Kmowles I.W. (ed.), Saito Y. (ed.)

By Kmowles I.W. (ed.), Saito Y. (ed.)

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If x(l — x) O2y + 2«/ = 0, prove that »(1 - x) On+2y + n(l - 2x) T>n+1y = (n + 1) (n - 2) Dny. 4. If ^ = sinh (m sinh _1 ic), and yn = dny/dxnf prove that (1 + x2)yn+2 + (2n + l)«y w + i + (n· - m2)yn = 0. 5. Prove that Bn+1(xy) = (n + 1) Dwy + zD w + 1 2/. By taking y — xn~x e 1 ^ prove, by induction, that e*/* D»(a*-iei/*) = ( _ i ) « _ . sin a; 6. r. to x and y as lim/(» Λ->0 + *,y)-/(«,y) Ä and lim/(*,» fc_>o +*)-/(*,») & respectively when these limits exist. r. r. t o x when y is t r e a t e d as a constant.

Volume II, Chapter I) is f(xy eloge /(*) = Before finding the derivative of log x we establish an important preliminary theorem or lemma. The relation y = f(x) implies that x is some function of y, g(y) say; treating y as the independent variable we may find dxjdy. , dy dx dx = 1. 17) 42 A COTJBSE OF M A T H E M A T I C S If y — log x, b y definition x = ey a n d hence dx/dy Therefore eqn. 17) gives = ey = x. 18) ——v(log 6 x) = — . 19) aï"*"»]-^ Examples, (i) D [log (x* + a*)] (ii) (iii) pxv (a* + a») D [log (a;2 + a2)w] = D [Λ log (a;2 -f a2)] D log a; + « 2nx (a;2 + a2) D [log (x + a) — log (s — a)] 1 a + a 1 x —a (iv) Express d2x/dy2 in terms of dy/dx, Since d2y/dx2.

Clearly y = s i n - 1 a? , s i n - 1 (^) = ±ηπ + (— l)n π / 6 where n is 0 or an integer. W e define t h e principal value of y = s i n - 1 x as t h a t value which lies in t h e range — \π < y < \π. Similarly t h e other inverse trigonometric functions are many-valued. We choose t h e principal values of u = c o s - 1 x a n d v = t a n - 1 x t o lie in t h e ranges 0 < u < π a n d — \π < v < ^π respectively. If y = s i n - 1 (xja), where a > 0 , t h e n # = a sin y. · . /) = ± / ( f t 2 — # 2 ) · .

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