Engineering Mathematics by Np Bali

AE51/AC51/AT51 Q.2

ENGINEERING MATHEMATICS-I

DEC 2013

a. If Z is a homogeneous function of degree n in x and y, show that ∂ 2z

∂ 2z ∂ 2z + y2 = n (n − 1) z ∂x∂y ∂x 2 ∂y 2 Answer: Page Number 221 of Text Book 1 x2

+ 2 xy

b. Use method of differentiation under integral sign to show that ∞

tan −1 ax

π

∫ x(1 + x 2 ) dx = 2 log(1 + a ), a ≥ 0 0

Answer:

4a 2 ay

Q.3

a. Change the order of integration and then evaluate

∫ 2∫ 0 y

dx dy

4a

Answer: Page Number 298-299 of Text Book 1 b. Find the volume bounded by the xy-plane, the cylinder x 2 + y 2 = 1 and the plane x+y+z = 3. Answer:

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AE51/AC51/AT51

Q.4

ENGINEERING MATHEMATICS-I

DEC 2013

 6 −2 2  a. Find the eigen values and eigen vectors of the matrix − 2 3 − 1  2 − 1 3 

Answer:

b. Determine the rank of the following matrices: 0 1 1 2 3  1 0   (i) 1 4 2 (ii)  3 1 2 6 5  1 1 Answer: Page Number 40 of Text Book 1

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− 3 − 1 1 1  0 2  −2 0 

2

AE51/AC51/AT51

ENGINEERING MATHEMATICS-I

DEC 2013

a. Use Regula-Falsi method to compute real root of xe x = 2 correct to three decimal places. Answer:

Q.5

b.

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Find by Runge-Kutta method of order four, an approximate value of y at dy y − x x = 0.2 for the equation = , y(0 ) = 1 . Take h = 0.2. dx y + x

3

AE51/AC51/AT51

ENGINEERING MATHEMATICS-I

DEC 2013

Answer:

Q.6a.

Solve the differential equation (x + 1)

dy + 1 = 2e − y dx

Answer:

b. Find the orthogonal trajectories of family of curves ay 2 = x 3 Answer:

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AE51/AC51/AT51

Q.7

ENGINEERING MATHEMATICS-I

a. Solve the differential equation

d2y dx

b. Solve the equation

d2y dx

2

+

2

−6

DEC 2013

dy + 25 y = e 2 x + x + Sinx dx

log x 1 dy = 12 x dx x2

Answer:

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AE51/AC51/AT51

Q.8

ENGINEERING MATHEMATICS-I

(

a. Obtain the series solution of 1 − x 2

DEC 2013

)dxd y2 − 2x dxdy + 2y = 0 2

Answer:

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AE51/AC51/AT51 b. Show that β(m, n ) =

ENGINEERING MATHEMATICS-I

DEC 2013

mn m+n

Answer: Page Number 854 of Text Book 1 Q.9 a. State and prove Rodrigue’s formula. Answer:

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AE51/AC51/AT51

ENGINEERING MATHEMATICS-I

b. Prove that J n +1 (n ) + J n −1 (n ) =

DEC 2013

2n J n (n ) x

Answer:

TEXT BOOKS 1. Higher Engineering Mathematics, Dr. B.S.Grewal, 40th edition 2007, Khanna publishers, Delhi. 2. Text book of Engineering Mathematics, N.P. Bali and Manish Goyal, 7th Edition 2007, Laxmi Publication (P) Ltd.

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