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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**Sample text**

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.