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By H.K. Dass

Offers with partial differentiation, a number of integrals, functionality of a fancy variable, detailed services, laplace transformation, advanced numbers, and records.

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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. We have, u = x log xy  1  u = x  . 1  = 1  log xy  x  x2  y    y  x  y 2  x x2  y . y x  y2 [From (1), (2), (3)] Ans.

X  x   y y  y  x       means   , means   .    r r  r   Example 19. If x = r cos , y = r sin , find  x   r  y (i)   (ii)   (iii)    r   x  y   r Solution. constant.    (iv)    y  x x  (i)   means the partial derivative of x with respect to r, keeping  as  r   x    = cos   r  x = r cos   y  (ii)   means the partial derivative of y with respect to , treating r as constant.   r  y  y = r sin    = r cos    r  r  (iii)   means the partial derivative of r with respect to x, treating y as constant.

0 x y x 2 y 2 2 2 4 15. Verify Euler’s theorem on homogeneous function when f (x, y, z) = 3x yz + 5xy z + 4z 2 14. If z = xy/(x + y), find the value of x Y 16. If u = x   X X2  2u x 2  Y     X  2 XY 2 z Ans. 0  2 xy   , prove by Euler’s theorem on homogeneous function that   2u  2u  Y 2 2 = 0. x y y 17. Given F (u) = V(x, y, z) where V is a homogeneous function of x, y, z of degree n, prove that x u u u F (u ) y z =n x y z F (u ) x y z 18. State and prove Euler’s theorem, and verify for u = y  z  x 19.

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