By Brian H. Chirgwin, Charles Plumpton
ISBN-10: 0080063888
ISBN-13: 9780080063881
Read Online or Download A Course of Mathematics for Engineers and Scientists. Volume 1 PDF
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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. -2a da; _ 1 di/ dy_ ' dx then by the chain rule. d^__d_ / 1 m dy2 ~ dy I dy \ dx d 2x •'•oV - / — da; I dy \~dx~ d'y da;2 da; dy dx We now find lim logo;.
N -f- βη + γη, where α, β and γ are the non-zero roots of the cubic equation xs -\- px2 -f qx -\- r = 0, show t h a t sn + psn_x + osn_2 -f rs TO _ 3 = 0. Hence express (i) a 4 -f β 4 + y 4 , (ii) a - 3 + β~ 3 + 7~z m terms of p, q, r. 10. By means of a sketch-graph, or otherwise, show t h a t the equation 2x3 — éx2 -{- 1 = 0 has two positive roots and one negative root. Calculate the negative root and the larger positive root correct to three significant figures. 11. By the substitution y = l/x find the equation in y whose roots are t h e reciprocals of the roots α, β, y of the equation xz -f- px2 + qx + T = 0 .
M2(m2 — 22) (m2 - 42) ... [m2 — (2p — 2) 2 ], /(2p + i)(0) = m(m2 - l2) (m2 - 32) ... [m2 — (2p - l) 2 ]. 58 A C O U R S E OF MATHEMATICS Exercises 2:7 x 4 1. If y = e~ cos x, prove that d i//da;4 + 4y = 0. 2. If y = (x3 - 3x2)e2x, find d«y/dxQ. 3. 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/.