A Textbook of Engineering Mathematics-I, 2nd Edition by H.S. Gangwar, Dr. Prabhakar Gupta

A Textbook of Engineering Mathematics-I, 2nd Edition by H.S. Gangwar, Dr. Prabhakar Gupta

By H.S. Gangwar, Dr. Prabhakar Gupta

Written for the scholars of BTech I yr of UP Technical college, Lucknow and different states, this booklet discusses intimately the recommendations and methods in Engineering arithmetic.

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Extra resources for A Textbook of Engineering Mathematics-I, 2nd Edition

Example text

2 2 x −y . 1 4 4. x 3 y − 3 tan–1 FG y IJ. H xK 2. Fx GH 5. b x + yg . 1 4 1 + y4 I Fx JK GH 1 5 1 I JK + y5 . xy 3. cos–1 6. cos–1 F xI GH y JK F xI GH y JK + cot–1 + cot–1 FG y IJ. H xK FG y IJ . H xK F x + y I , show that x ∂u + y ∂u = 3. , 2000) GH x + y JK ∂y ∂x ∂u 9 ∂u I F I F 8. U. (AG), 2005] 20 ∂x F xI F xI ∂z ∂z 9. If z = x y sin G y J + log x – log y, show that x +y = 6x y sin G y J . H K H K ∂x ∂y 4 4 7. U. , 2003] ∂ ∂ ∂ u u u 10. If u = x3 + y3 + z3 + 3xyz; show that x +y +z = 3u. ∂x ∂y ∂z 11.

X, we get x or ∂u ∂ 2u ∂u ∂ 2u + + y = 2 cos 2u . t. (iii) Adding (ii) and (iii), we get x2 ∂ 2u ∂ 2u ∂ 2u 2 = ( 2cos 2u – 1) 2 + 2xy ∂x∂y + y ∂x ∂y2 F x ∂u + y ∂u I GH ∂x ∂y JK = (2cos 2u – 1) sin 2u, (from (i)) = (2sin 2u cos 2u – sin 2u) = sin 4u – sin 2u = 2 cos FG 4u + 2uIJ H 2 K FG 4u − 2uIJ . H 2 K . cos ∂ 2u ∂ 2u 2 Hence, x + 2xy +y ∂y2 = 2 cos 3u · cos u. ∂x∂y ∂x2 2 ∂ 2u Example 8. If z = xm f x2 Sol. (i) Now, u is homogeneous function of degree m. (iii) Adding (ii) and (iii), we get ∂ ∂2 ∂2 2 x 2 (u + v) + 2xy ∂x∂y (u + v) + y ∂y 2 (u + v) = m (m – 1) u + n (n – 1) v ∂x 2 ⇒ x2 ∂ 2z ∂2z ∂ 2z 2 + 2xy + y ∂y2 = m (m – 1) u + n (n – 1) v (As z = u + v).

U. , 2003] ∂ ∂ ∂ u u u 10. If u = x3 + y3 + z3 + 3xyz; show that x +y +z = 3u. ∂x ∂y ∂z 11. If u = cos–1 LM x − y OP , prove that x Nx + y Q ∂u ∂u + y ∂y = 0. ∂x 12. If u = log [(x2 + y2)/(x + y)], prove that x 13. If u = sin–1 {(x2 + y2)/(x + y)}, show that x 14. If u = sin–1 {(x + y)/( x + ∂u ∂u + y ∂y = 1. , 2008) ∂u ∂u + y ∂y = tan u. ∂x y )}. Show that x ∂u 1 ∂u + y ∂y = tan u. 2 ∂x ∂u 1 ∂u + y ∂y = sin 2u. 2 ∂x 16. If u is a homogeneous function of degree n, show that 15. If u = tan–1 [(x2 + y2)/(x + y)], then prove that x ∂ 2u ∂ 2u ∂u .