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By H.K. Dass

Bargains with partial differentiation, a number of integrals, functionality of a fancy variable, particular features, laplace transformation, complicated numbers, and facts.

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If w = f (u, v), where u = x + y and v = x – y, show that  2 dx dy du 2. If z = sin–1 (x – y), x = 3 t, y = 4 t3; show that 4. If u = xey z, where y = du 3 a 2  x 2 , z = sin x. Find d x 5. If u = x 2 + y2 + z2 – 2 xyz = 1, show that [Hint. du =  x2 Ans. eyz 1   3x cot  y  dx dy dz   0 2 2 1 x 1 y 1 z2  x   u u u dx  dy  dz  0 x y z = 2 (x – yz) dx + 2 (y – zx) dy + 2 (z – xy) dz = 0 But x2 + y2 + z2 – 2 xyz = 1,  y2 – 2xyz = 1 – x2 – z2 2 2 2 2 2 2 2 y – 2xyz + x z = 1 + x z – x – z  (y – xz)2 = (1 – x2) (1 – z2)] 2 2 6.

2u x 2  2 xy  2u  2u  y 2 2 = g(u)[g(u) – 1] xy y sin 2u = g(u)  2u  2u  y 2 2 = sin 2u (2 cos 2u – 1) = 2 sin 2u cos 2u – sin 2u xy y = sin 4u – sin 2u = 2 cos 3u sin u Proved.  xy  1  Example 29. If u = sin   x  y  2  2u  sin u cos 2u 2  u  y  . 2 2 xy x y 4 cos3 u x y 1 Solution. We have, u = sin x y y  x 1   x y x    x1/ 2  ( x) Let z = sin u =   x y y x 1   x  z = f (u) = sin u 1 z is a homogeneous function of degree . 2 By Euler’s deduction I u u f (u ) u u 1 sin u x y y = n   x = x y f (u ) x y 2 cos u u u 1 x y tan u = x y 2 Prove that x 2  2u  2 xy Created with Print2PDF.

F  Example 44. If f (x, y) = 0 and  (y, z) = 0, show that  y .  z . d x   x .  y . Solution. t. (2) f f dy  . t. ‘y’, we get 0 =   d z  .  y z d y f dy x =  f dx y  dz y  =   dy z Multiplying (3) and (4), we get   f      x  y  dy dz      =  dx dy   f      y  z      f  d z  f   . = . (4)  f   dz x y =  f  dx   y z Proved. P. , Dec. 2005, Com. 2002) Example 45. If u = x log xy where x3 + y3 + 3 xy = 1. Find Solution.

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