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GUJARAT TECHNOLOGICAL UNIVERSITY

CIVIL & INFRASTRUCTURE ENGINEERING

ADVANCE ENGINEERING MATHEMATICS

SUBJECT CODE: 2130002

B.E. 3rd

SEMESTER

Type of course: Engineering Mathematics

Prerequisite: The course follows from Calculus, Linear algebra

Rationale: Mathematics is a language of Science and Engineering

Teaching and Examination Scheme:

Content:

Sr.

No. Topics

Teaching

Hrs.

Module

Weightage

1

Introduction to Some Special Functions: Gamma function, Beta function, Bessel function, Error function and

complementary Error function, Heaviside’s function, pulse unit height

and duration function, Sinusoidal Pulse function, Rectangle function,

Gate function, Dirac’s Delta function, Signum function, Saw tooth wave

function, Triangular wave function, Halfwave rectified sinusoidal

function, Full rectified sine wave, Square wave function.

02

4

2

Fourier Series and Fourier integral: Periodic function, Trigonometric series, Fourier series, Functions of any

period, Even and odd functions, Half-range Expansion, Forced

oscillations, Fourier integral

05

10

3

Ordinary Differential Equations and Applications: First order differential equations: basic concepts, Geometric meaning of

y’ = f(x,y) Direction fields, Exact differential equations, Integrating

factor, Linear differential equations, Bernoulli equations, Modeling ,

Orthogonal trajectories of curves.Linear differential equations of second

and higher order: Homogeneous linear differential equations of second

order, Modeling: Free Oscillations, Euler- Cauchy Equations,

Wronskian, Non homogeneous equations, Solution by undetermined

coefficients, Solution by variation of parameters, Modeling: free

Oscillations resonance and Electric circuits, Higher order linear

differential equations, Higher order homogeneous with constant

coefficient, Higher order non homogeneous equations. Solution by

[1/f(D)] r(x) method for finding particular integral.

11

20

Teaching Scheme Credits Examination Marks Total

Marks L T P C Theory Marks Practical Marks

ESE

(E)

PA (M) PA (V) PA

(I) PA ALA ESE OEP

3 2 0 5 70 20 10 30 0 20 150

4

Series Solution of Differential Equations: Power series method, Theory of power series methods, Frobenius

method.

03

6

5

Laplace Transforms and Applications: Definition of the Laplace transform, Inverse Laplace transform,

Linearity, Shifting theorem, Transforms of derivatives and integrals

Differential equations, Unit step function Second shifting theorem,

09

15

Dirac’s delta function, Differentiation and integration of transforms,

Convolution and integral equations, Partial fraction differential

equations, Systems of differential equations

6

Partial Differential Equations and Applications:

Formation PDEs, Solution of Partial Differential equations f(x,y,z,p,q) = 0, Nonlinear PDEs first order, Some standard forms of nonlinear

PDE, Linear PDEs with constant coefficients,Equations reducible to

Homogeneous linear form, Classification of second order linear

PDEs.Separation of variables use of Fourier series, D’Alembert’s

solution of the wave equation,Heat equation: Solution by Fourier series

and Fourier integral

12

15

Reference Books:

1. Advanced Engineering Mathematics (8th Edition), by E. Kreyszig, Wiley-India (2007). 2. Engineering Mathematics Vol 2, by Baburam, Pearson

3. W. E. Boyce and R. DiPrima, Elementary Differential Equations (8th Edition), John Wiley (2005)

4. R. V. Churchill and J. W. Brown, Fourier series and boundary value problems (7th Edition),

McGraw-Hill (2006).

5. T.M.Apostol, Calculus , Volume-2 ( 2nd Edition ), Wiley Eastern , 1980

Course Outcome:

After learning the course the students should be able to

1. Fourier Series and Fourier Integral

o Identify functions that are periodic. Determine their periods.

o Find the Fourier series for a function defined on a closed interval.

o Find the Fourier series for a periodic function.

o Recall and apply the convergence theorem for Fourier series.

o Determine whether a given function is even, odd or neither.

o Sketch the even and odd extensions of a function defined on the interval [0,L].

o Find the Fourier sine and cosine series for the function defined on [0,L]

2. Ordinary Differential Equations and Their Applications

o Model physical processes using differential equations.

o Solve basic initial value problems, obtain explicit solutions if possible.

o Characterize the solutions of a differential equation with respect to initial values.

o Use the solution of an initial value problem to answer questions about a physical system. o Determine the order of an ordinary differential equation. Classify an ordinary differential equation as

linear or nonlinear.

o Verify solutions to ordinary differential equations.

o Identify and solve first order linear equations.

o Analyze the behavior of solutions.

o Analyze the models to answer questions about the physical system modeled.

o Recall and apply the existence and uniqueness theorem for first order linear differential equations.

o Identify whether or not a differential equation is exact.

o Use integrating factors to convert a differential equation to an exact equation and then solve. o Solve second order linear differential equations with constant coefficients that have a characteristic equation

with real and distinct roots.

o Describe the behavior of solutions. o Recall and verify the principal of superposition for solutions of second order linear differential

equations. o Evaluate the Wronskian of two functions.

1

Seat No.: ________ Enrolment No.___________

GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTER–III (NEW) - EXAMINATION – SUMMER 2017

Subject Code: 2130002 Date: 25/05/2017 Subject Name: Advanced Engineering Mathematics Time: 10:30 AM to 01:30 PM Total Marks: 70 Instructions:

1. Attempt all questions. 2. Make suitable assumptions wherever necessary. 3. Figures to the right indicate full marks.

MARKS

Q.1 Short Questions 14

1 What are the order and the degree of the differential

equation xyy cos33"2 .

2 What is the integrating factor of the linear differential

equation: 2)/1(' xyxy

3 Is the differential equation 0)2( dyeydxyexx is

exact? Justify.

4 Solve: 010'11" yyy .

5 Find particular integral of : xeyy

2''"

6 If xexccy )( 21 is a complementary function of a

second order differential equation, find the Wronskian

).,( 21 yyW

7 Find the value of

2

7

8 What is the value of the Fourier coefficients a0 and bn for

.11,)(2

xxxf

9 Find 33 teL

10 Find

)9(

14

22

1

ss

L

11 Find the singular point of the differential equation

0)1('2")1(2

ynnxyyx

12 Obtain the general integral of 0

3

3

x

z

13 Obtain the general integral of zqp

14 State the relationship between beta and gamma function.

Q.2 (a) Solve: 02)3(22

xydydxyx 03

(b) Solve: 0)0(,2sin)(tan yxyx

dx

dy

04

(c) dxdDwherexeyD

x/,)16(

424 07

OR

(c) Use the method of variation of parameters to find the 07

2

general solution of x

eyyy

x2

4'4"

Q.3 (a) Find half range sine series of xxxf 0,)(3 03

(b) Find the Fourier integral representation of the function

2

2

,0

,2)(

x

xxf

04

(c) Find the Fourier series expansion for the 2 - periodic

function 2

)( xxxf in the interval x and show

that 12

...

4

1

3

1

2

1

1

12

2222

07

OR

Q.3 (a) Discuss about ordinary point, singular point, regular

singular point and irregular singular point for the

differential equation: 07')1(3")1(3

xyyxyxx

03

(b) Use the method of undetermined coefficients to solve the

differential equation 229" xyy

04

(c) Find the series solution of 0'")1(2

xyxyyx

about 00 x .

07

Q.4 (a) Solve: dxdDwherexeyDx

/,)1(2

03

(b) Solve: )sin(ln

2

22

xydx

dyx

dx

ydx

04

(c) Use Laplace Transform to solve the following initial

value problem:

6)0(',2)0(,122'3"2

yyeyyyt

07

OR

Q.4 (a) Obtain teLt 22

sin 03

(b) Find

258

7

2

1

ss

sL

04

(c) Using Convolution theorem, obtain

22

1

)4(

1

s

L 07

Q.5 (a) Find the Laplace Transform of tett

2cos4

03

(b) Form the partial differential equation from the following:

1) ctbyaxz 2)

y

xfz

04

(c) Using the method of separation of variables solve,

ut

u

x

u

2 where

xexu

36)0,(

07

OR

Q.5 (a) Obtain the solution of the partial differential equation:

,yxqp 22

where y

zq

x

zp

,

03

(b) Solve: )yz(xxyqpy 2

2 ,where

y

zq,

x

zp

04

(c) Find the solution of the wave equation 07

3

Lxucu xxtt 0,2

satisfying the conditions:

LxL

xxuxutLutu t 0,)0,(,0)0,(,0),(),0(

*************

1


GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTER–III(New) • EXAMINATION – WINTER 2016

Subject Code:2130002 Date:30/12/2016

Subject Name:Advanced Engineering Mathematics

Time: 10:30 AM to 01:30 PM Total Marks: 70 Instructions:

1. Attempt all questions.

2. Make suitable assumptions wherever necessary.

3. Figures to the right indicate full marks.

MARKS

Q.1 Answer the following one mark questions 14

1 Find ⌈(1

2).

2 State relation between beta and gamma function.

3 Define Heaviside’s unit step function.

4 Define Laplace transform of f (t), t ≥ 0.

5 Find Laplace transform of 𝑡−1

2 .

6 Find L { 𝑠𝑖𝑛𝑎𝑡

𝑡 }, given that L {

𝑠𝑖𝑛𝑡

𝑡 } = 𝑡𝑎𝑛−1{

1

𝑠 }.

7 Find the continuous extension of the function f(x) = 𝑥2+𝑥−2

𝑥2−1 to x = 1

8 Is the function f(x) = 1

𝑥 continuous on [-1, 1] ? Give reason.

9 Solve 𝑑𝑦

𝑑𝑥 = 𝑒3𝑥−2𝑦 + 𝑥2𝑒−2𝑦.

10 Give the differential equation of the orthogonal trajectory of the

family of circles 𝑥2 + 𝑦2 = 𝑎2.

11 Find the Wronskian of the two function sin2x and cos2x .

12 Solve ( 𝐷2 + 6𝐷 + 9) x = 0; D = 𝑑

𝑑𝑡.

13 To solve heat equation 𝜕𝑢

𝜕𝑡= 𝑐2 𝜕2𝑢

𝜕𝑥2 how many initial and

boundary conditions are required.

14 Form the partial differential equations from z = f(x + at) + g(x –at).

Q.2 (a) Solve : (x+1)𝑑𝑦

𝑑𝑥− 𝑦 = 𝑒3𝑥(𝑥 + 1)2. 03

(b) Solve : 𝑑𝑦

𝑑𝑥+

𝑦𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑦 +𝑦

𝑠𝑖𝑛𝑥 + 𝑥𝑐𝑜𝑠𝑦 + 𝑥= 0 04

(c) Find the series solution of 𝑑2𝑦

𝑑𝑥2 + 𝑥𝑦 = 0. 07

OR

(c) Find the general solution of 2𝑥2𝑦′′ + 𝑥𝑦′ + (𝑥2 − 1)𝑦 = 0 by using frobenius method.

07

Q.3 (a) Solve : (𝐷3 − 3𝐷2 + 9𝐷 − 27)y = cos3x. 03

(b) Solve : 𝑥2 𝑑2𝑦

𝑑𝑥2 + 4𝑥𝑑𝑦

𝑑𝑥+ 2𝑦 = 𝑥2 sin(𝑙𝑛𝑥). 04

(c) (i) Solve : 𝜕3𝑧

𝜕𝑥3 − 2𝜕3𝑧

𝜕𝑥2𝜕𝑦= 2𝑒2𝑥.

(ii) find the general solution to the partial differential equation

03

04

2

(𝑥2 − 𝑦2 − 𝑧2)𝑝 + 2𝑥𝑦𝑞 = 2𝑥𝑧.

OR

Q.3 (a) Solve : (𝐷3 − 𝐷)𝑦 = 𝑥3 . 03

(b) Find the solution of 𝑦′′ − 3𝑦′ + 2y = 𝑒𝑥 , using the method of

variation of parameters.

04

(c) Solve 𝑥𝜕𝑢

𝜕𝑥 - 2y

𝜕𝑢

𝜕𝑦= 0 using method of separation of variables. 07

Q.4 (a) Find the Fourier cosine integral of 𝑓(𝑥) = 𝑒−𝑘𝑥, 𝑥 > 0, 𝑘 > 0 03

(b) Express f(x) = |x|, −𝜋 < 𝑥 < 𝜋 as fouries series. 04

(c) Find Fourier Series for the function f(x) given by

𝑓(𝑥) = {1 +

2𝑥

𝜋; −𝜋 ≤ 𝑥 ≤ 0

1 −2𝑥

𝜋 ; 0 ≤ 𝑥 ≤ 𝜋

Hence deduce that 1

12 + 1

32 +1

52 + … . =𝜋2

8 .

07

OR

Q.4 (a) Obtain the Fourier Series of periodic function function

f(x) =2x, -1 < x < 1, p = 2L =2

03

(b) Show that ∫𝑠𝑖𝑛𝜆𝑐𝑜𝑠𝜆

𝜆

∞

0𝑑𝜆 = 0, if x ˃ 1. 04

(c) Expand f(x) in Fourier series in the interval (0, 2𝜋) if

𝑓(𝑥) = {−𝜋 ; 0 < 𝑥 < 𝜋

𝑥 − 𝜋 ; 𝜋 < 𝑥 < 2𝜋

and hence show that ∑1

(2𝑟+1)2 =𝜋2

8.∞

𝑟=0

07

Q.5 (a) Find L{∫ 𝑒𝑡𝑡

0

𝑠𝑖𝑛𝑡

𝑡𝑑𝑡}. 03

(b) Find 𝐿−1{ 2𝑠2−1

(𝑠2+1)(𝑠2+4) }. 04

(c) Solve initial value problem : 𝑦′′ − 3𝑦′ + 2y = 4t + 𝑒3𝑡 ,y(0) = 1 and

𝑦′(0) = −1 ,using Laplace transform.

07

OR

Q.5 (a) Find L {tsin3tcos2t}. 03

(b) Find 𝐿−1{ 𝑒−3𝑠

𝑠2+8𝑠+25 }. 04

(c) State the convolution theorem and apply it to evaluate 𝐿−1 { 𝑠

(𝑠2+𝑎2)2 }. 07

*************

1



BE - SEMESTER–III(New) EXAMINATION – SUMMER 2016


Subject Name:Advanced Engineering Mathematics Time:10:30 AM to 01:30 PM Total Marks: 70 Instructions:


2. Make suitable assumptions wherever necessary. 3. Figures to the right indicate full marks.

Q.1 Answer the following one mark each questions : 14

1 Integreating factor of the differential equation

𝑑𝑥

𝑑𝑦+

3𝑥

𝑦=

1

𝑦2 is _________

Type equation here.

2 The general solution of the differential equation 𝑑𝑦

𝑑𝑥+

𝑦

𝑥

=tan2x____________.

3 The orthogonal trajectory of the family of curve 𝑥2 +𝑦2 = 𝑐2 is _________

4 Particular integral of (𝐷2 + 4)𝑦 = cos 2𝑥 is _________

5 X=0 is a regular singular point of 2𝑥2𝑦′′ + 3𝑥𝑦′(𝑥2 − 4)𝑦 = 0 say true or false.

6 The solution of

(𝑦 − 𝑧)𝑝 + (𝑧 − 𝑥)𝑞 = 𝑥 − 𝑦 𝑖𝑠_________

7 State the type ,order and degree of differential equation

(𝑑𝑥

𝑑𝑦)2 + 5𝑦

1

3 = x is __________

8 Solve (D+𝐷′)z= cos x

9 Is the partial differential equation

2𝜕2𝑢

𝜕𝑥2+4𝜕2𝑢

𝜕𝑥𝜕𝑦+ 3

𝜕2𝑢

𝜕𝑦2 = 6 elliptic?

10 𝐿−1 (

1

(𝑠 + 𝑎)2) = ____________

11 If f(t) is a periodic function with period t then

L[𝑓(𝑡)] = _________

12 Laplace transform of f(t) is defined for +ve and –ve

values of t. Say true or false.

13 State Duplication (Legendre) formula.

14 Find B ( 9 ,

2

7

2 )

Q.2 (a) Solve : 9y 𝑦 ˊ + 4𝑥 = 0 03

2

(b) Solve : 𝑑𝑦

𝑑𝑥+ 𝑦 cot 𝑥 = 2 cos 𝑥 04

(c) Find series solution of 𝑦 ˊˊ + 𝑥𝑦 = 0 07

OR

(c) Determine the value of (a) J1

2 (𝑥) (𝑏) J3

2 (𝑥) 07

Q.3 (a) Solve (𝐷2 + 9)𝑦 = 2sin 3𝑥 + cos 3𝑥 03

(b) Solve 𝑦′′ + 4𝑦′ = 8𝑥2 by the method of

undetermined coefficients.

04

(c) (i) Solve 𝑥2𝑝 + 𝑦2𝑞 = 𝑧2

(ii) Solve by charpit’s method px+qy = pq

07

OR

Q.3 (a) Solve 𝑦′′ + 4𝑦′ + 4 = 0 , 𝑦(0) = 1 , 𝑦′(0) = 1 03

(b) Find the solution of 𝑦′′ + 𝑎2𝑦′ = tan 𝑎𝑥 , by the

method of variation of parameters.

04

(c) Solve the equation ux = 2ut + u given u(x,0)= 4𝑒−4𝑥 by

the method of separation of variable. 07

Q.4 (a) Find the fourier transform of the function f(x) = 𝑒−𝑎𝑥2

03

(b) Obtain fourier series to represent f(x) =x2 in the interval

-𝜋 < 𝑥 < 𝜋 .Deduce that ∑1

𝑛2∞𝑛=1 =

𝜋2

6

04

(c) Find Half-Range cosine series for

F(x) = kx , 0≤ 𝑥 ≤𝑙

2

= k(𝑙-x ) , 𝑙

2 ≤ 𝑥 ≤ 𝑙

Also prove that ∑1

(2𝑛−1)2∞𝑛=1 =

𝜋2

8

07

OR

Q.4 (a) Expres the function

F(x)= 2 , |x| < 2

= 0 , |x| > 2 as Fourier integral.

03

(b) Find the fourier series expansion of the function

F(x) = -𝜋 − 𝜋 < 𝑥 < 0

= x 0 < 𝑥 < 𝜋

04

(c) Find fourier series to represent the function

F(x) = 2x-x2 in 0 < 𝑥 < 3

07

Q.5 (a) 𝐹𝑖𝑛𝑑 𝐿−1 {1

(𝑠+√2)(𝑠−√3)}

03

(b) Find the laplace transform of

(i) 𝑐𝑜𝑠𝑎𝑡−𝑐𝑜𝑠𝑏𝑡

𝑡

(ii) tsinat

04

(c) State convolution theorem and use to it evaluate

𝐿−1 {1

(𝑠2+𝑎2)2}

07

3

OR

Q.5 (a) L {𝑡2 cos ℎ3𝑡} 03

(b) Find 𝐿−1 {1

𝑠4−81} 04

(c) Solve the equation 𝑦′′ − 3𝑦′ + 2𝑦 = 4𝑡 + 𝑒3𝑡,when

y(0)=1 , 𝑦′(0) = −1

07

*************

1


GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTER–III (New) EXAMINATION – WINTER 2015


Subject Name: Advanced Engineering Mathematics

Time: 2:30pm to 5:30pm Total Marks: 70 Instructions:




Q.1 Answer the following one mark each questions: 14

1 Find Γ (13

2)

2 State relationship between beta and gamma functions.

3 Represent graphically the given saw-tooth function 𝑓(𝑥) = 2𝑥, 0 ≤𝑥 < 2 and 𝑓(𝑥 + 2) = 𝑓(𝑥) for all 𝑥.

4 For a periodic function 𝑓 with fundamental period p, state the formula to

find Laplace transform of 𝑓.

5 Find 𝐿(𝑒−3𝑡𝑓(𝑡)), if 𝐿(𝑓(𝑡)) =𝑠

(𝑠−3)2.

6 Find 𝐿[(2𝑡 − 1)2].

7 Find the extension of the function 𝑓(𝑥) = 𝑥 + 1, define over (0,1] to [−1, 1] − {0} which is an odd function.

8 Is the function 𝑓(𝑥) = {

𝑥, 0 ≤ 𝑥 ≤ 2

𝑥2, 2 < 𝑥 ≤ 4; continuous on [0,4]? Give

reason.

9 Is the differential equation 𝑑𝑦

𝑑𝑥=

𝑦

𝑥 exact? Give reason.

10 Give the differential equation of the orthogonal trajectory to the equation

𝑦 = 𝑐𝑥2.

11 If 𝑦 = 𝑐1𝑦1 + 𝑐2𝑦2 = 𝑒𝑥(𝑐1 cos 𝑥 + 𝑐2 sin 𝑥) is a complementary

function of a second order differential equation, find the Wronskian

𝑊(𝑦1, 𝑦2).

12 Solve (𝐷2 + 𝐷 + 1)𝑦 = 0; where 𝐷 =𝑑

𝑑𝑡.

13 Is 𝑢(𝑡, 𝑥) = 50𝑒(𝑡−𝑥) 2⁄ , a solution to 𝜕𝑢

𝜕𝑡=

𝜕𝑢

𝜕𝑥+ 𝑢?

14 Give an example of a first order partial differential equation of Clairaut’s

form.

Q.2 (a) Solve: 𝑑𝑦

𝑑𝑥=

𝑥2−𝑥−𝑦2

2𝑥𝑦. 03

2

(b) Solve: 𝑑𝑦

𝑑𝑥+

1

𝑥𝑦 = 𝑥3𝑦3. 04

(c) Find the series solution of (𝑥 − 2)𝑑2𝑦

𝑑𝑥2− 𝑥2 𝑑𝑦

𝑑𝑥+ 9𝑦 = 0 about 𝑥0 = 0. 07

OR

(c) Explain regular-singular point of a second order differential equation and

find the roots of the indicial equation to 𝑥2𝑦′′ + 𝑥𝑦′ − (2 − 𝑥)𝑦 = 0.

07

Q.3 (a) Find the complete solution of 𝑑3𝑦

𝑑𝑥3 + 8𝑦 = cosh(2𝑥). 03

(b) Find solution of 𝑑2𝑦

𝑑𝑥2 + 9𝑦 = tan 3𝑥, using the method of variation of

parameters.

04

(c) Using separable variable technique find the acceptable general solution to

the one-dimensional heat equation 𝜕𝑢

𝜕𝑡= 𝑐2

𝜕2𝑢

𝜕𝑥2 and find the solution

satisfying the conditions 𝑢(0, 𝑡) = 𝑢(𝜋, 𝑡) = 0 for 𝑡 > 0 and 𝑢(𝑥, 0) =𝜋 − 𝑥, 0 < 𝑥 < 𝜋.

07

OR

Q.3 (a) Solve completely, the differential equation 𝑑2𝑦

𝑑𝑥2 − 6𝑑𝑦

𝑑𝑥+ 9𝑦 = cos(2𝑥) sin 𝑥.

03

(b) Solve completely the differential equation

𝑥2 𝑑2𝑦

𝑑𝑥2 − 6𝑥𝑑𝑦

𝑑𝑥+ 6𝑦 = 𝑥−3 log 𝑥.

04

(c) (i) Form the partial differential equation for the equation (𝑥 − 𝑎)(𝑦 −𝑏) − 𝑧2 = 𝑥2 + 𝑦2.

(ii) Find the general solution to the partial differential equation 𝑥𝑝 +𝑦𝑞 = 𝑥 − 𝑦.

07

Q.4 (a) Find the Fourier cosine integral of 𝑓(𝑥) =𝜋

2𝑒−𝑥, 𝑥 ≥ 0. 03

(b) For the function 𝑓(𝑥) = cos 2𝑥, find its Fourier sine series over [0, 𝜋]. 04

(c) For the function 𝑓(𝑥) = {

𝑥; 0 ≤ 𝑥 ≤ 2 4 − 𝑥; 2 ≤ 𝑥 ≤ 4

, find its Fourier series.

Hence show that 1

12 +1

32 +1

52 + ⋯ =𝜋2

16.

07

OR

Q.4 (a) Find the Fourier cosine series of 𝑓(𝑥) = 𝑒−𝑥, where 0 ≤ 𝑥 ≤ 𝜋. 03

(b) Show that ∫𝜆3 sin 𝜆𝑥

𝜆4+4

∞

0𝑑𝜆 =

𝜋

2𝑒−𝑥 cos 𝑥, 𝑥 > 0. 04

(c) Is the function 𝑓(𝑥) = 𝑥 + |𝑥|, -𝜋 ≤ 𝑥 ≤ 𝜋 even or odd? Find its Fourier

series over the interval mentioned.

07

Q.5 (a) Find 𝐿 {∫ 𝑒𝑢(𝑢 + sin 𝑢)𝑑𝑢𝑡

0}. 03

(b) Find 𝐿−1 {1

𝑠(𝑠2−3𝑠+3)}. 04

(c) Solve the initial value problem: 𝑦′′ − 2𝑦′ = 𝑒𝑡 sin 𝑡, 𝑦(0) = 𝑦′(0) = 0,

using Laplace transform.

07

OR

Q.5 (a) Find 𝐿{𝑡(sin 𝑡 − 𝑡 cos 𝑡 )}. 03

(b) Find 𝐿−1 {𝑒−2𝑠

(𝑠2+2)(𝑠2−3)}. 04

(c) State the convolution theorem and verify it for 𝑓(𝑡) = 𝑡 and 𝑔(𝑡) = 𝑒2𝑡. 07

**********

1


GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTER– III (NEW)EXAMINATION – SUMMER 2015

Subject Code:2130002 Date: 06/06/2015

Subject Name:Advanced Engineering Mathematics

Time: 02.30pm-05.30pm Total Marks: 70 Instructions:




Q.1

(a) (1) Solve the differential equation2

1

x

e

xdx

dy y

. 04

(2) Solve the differential equation . 0)2( dyeydxye xx

03

(b)

Find the series solution of 09'")1( 2 yxyyx . 07

Q.2 (a) (1)Solve the differential equation using the method of variation of parameter

xyy 3sec 9 . 04

(2) Solve the differential equationxeyDD 10)12( 2 . 03

(b) Using the method of separation of variables, solve

xexuut

u

x

u 36)0,(;2

. 07

OR

(b) Find the series solution of 0;0)1()1(2 0 xyyxyxx 07

Q.3 (a) Find the Fourier Series for

xx

xxxf

0;

0;)( 07

(b) (1) Find the Half range Cosine Series for .10;)1()( 2 xxxf 04

(2) Find the Fourier sine series for .0;)( xexf x 03

OR

Q.3 (a) Find the Fourier Series for

xx

xxf

0;

0;)( . 07

(b) (1) Find the Fourier cosine series for .0;)( 2 xxxf 04

(2) Find the Fourier sine series for 10;2)( xxxf . 03

Q.4 (a) (1) Prove that (i) as

aseL at

;

1)( (ii)

22)(sinh

as

aatL

. 04

(2) Find the Laplace transform of tt 2sin . 03

(b) (1) Using convolution theorem, obtain the value of

4

12

1

ssL . 04

(2) Find the inverse Laplace transform of

32

1

ss. 03

OR

Q.4 (a) Solve the initial value problem using Laplace transform:

00,10,23 yyeyyy t.

07

(b) (1) Find the Laplace transform of

tt

ttf

;sin

0;0. 04

2

(2) Evaluatetet . 03

Q.5 (a) Using Fourier integral representation prove that

0

2

0

02

00

1

sincos

xife

xif

xif

dxx

x

. 07

(b) (1) Form the partial differential equation by eliminating the arbitrary functions from

0, 222 zyxzyxf . 04

(2) Solve the following partial differential equation xyqzxpyz .

03

OR

Q.5 (a) A homogeneous rod of conducting material of length 100 cm has its ends kept

at zero temperature and the temperature initially is

10050;100

500;)0,(

xx

xxxu

Find the temperature ),( txu at any time.

07

(b) (1) Solve .23

2

22

2

2

yxy

z

yx

z

x

z

04

(2) Solve 22 yqxp . 03

*************

1


GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTER–III • EXAMINATION – SUMMER • 2014

Subject Code: 130002 Date: 02-06-2014


Time: 02.30 pm - 05.30 pm Total Marks: 70 Instructions:




Q.1 (a) Solve 23 )1()1( xey

dx

dyx x

03

( (b) .0)1(,0])1([ ydyexdxeex yyx 04

(c) Determine the series solution for the differential equation 0'' yy about .00 x 07

Q.2 (a) (i) Solve

xeyDD )65( 2 . 03

(ii) Solve xy

dx

yd2tan4

2

2

by method of variation of parameter. 04

(b) Solve 02

2

2

y

z

x

z

x

z by the method of separation of variable.

07

OR

(b) Solve in series the equation 0

2

2

xydx

yd.

07

Q.3 (a) Find the Fourier series for 20,)( xexf x . 07

(b) Find the Fourier series expansion for

x

x

xxfifxf

0

0

,

,)(,)(

Deduce that 8

...........5

1

3

1

1

1 2

222

.

07

OR

Q.3 (a) Find the Fourier series expansions of

(i) )()2(,)( xfxfxxxf

(ii) lxlxxf 2)( .

07

(b) Express xxf )( as a

(i) half range sine series in 0 < x <2

(ii) half range cosine series in 0 < x < 2.

07

Q.4 (a)

(1) Prove that .)sinh(22

asforas

aatL

03

(2) Find the Laplace transforms of (i) sin2tsin3t (ii) )5sin35cos2(3 tte t 04

(b) Evaluate : (i) })1(ln{

2

21

s

wL

(ii) })52)(1(

35{

2

1

sss

sL

07

OR

Q.4 (a) Apply convolution theorem to evaluate

222

1

)( as

sL

03

(b) Use Laplace transform method to solve atkyay sin2'' 04

2

(c) If )())(( sftfL and if

t

tfL

)(exists then prove that

.)()(

dssft

tfL

s

Also find

t

tL

2sin

07

Q.5 (a) (1) Form partial differential equation of ,0),( 222 zyxzyxf where

f is an arbitary function.

03

(2) Solve xzxyqpzyx 22)( 222 04

(b) (1) Solve qzqp )1(

(2) Solve yxqp 22

07

OR

Q.5 (a) Define following terms (i) Beta function (ii) Sinusoidal function 04

(b) Form partial differential equation 22 )3()2( yxz 03

(c) Find the Fourier integral representation of the function

2,0

2,2)(

x

xxf

07

*************

1


GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTER–III (OLD) - EXAMINATION – SUMMER 2017

Subject Code: 130002 Date: 29/05/2017 Subject Name: Advanced Engineering Mathematics Time: 10:30 AM to 01:30 PM Total Marks: 70 Instructions:


Q.1 (a) 1) Solve (𝑒𝑦 − 𝑦𝑒𝑥)𝑑𝑥 + (𝑥𝑒𝑦 − 𝑒𝑥)𝑑𝑦 = 0

2) Solve 𝑦′ + (𝑥 + 1)𝑦 = 𝑒𝑥2𝑦3, 𝑦(0) = 0.5

03

04

(b) Find the series solution of 𝑦′′ + 𝑦 = 0. 07

Q.2 (a) Using the method of separation of variables solve 2

𝜕𝑢

𝜕𝑥=

𝜕𝑢

𝜕𝑡+ 𝑢,

𝑢(𝑥, 0) = 4𝑒−3𝑥.

07

(b) Find the series solution of the differential equation 3𝑥𝑦′′ + 2𝑦′ + 𝑦 = 0 07

OR

(b) 1) Solve 𝑦′′′ + 𝑦′ = 0

2) Solve 𝑦′′′ + 3𝑦′′ + 3𝑦′ + 𝑦 = 30𝑒−𝑥 by the method of undetermined

coefficients.

03

04

Q.3 (a) Find the Fourier series of 𝑓(𝑥) = 𝜋 − |𝑥|, −𝜋 < 𝑥 < 𝜋, 𝑓(𝑥 + 2𝜋) = 𝑓(𝑥). 07

(b) Find the Fourier series of 𝑓(𝑥) = 𝑒−4𝑥, −𝜋 < 𝑥 < 𝜋, 𝑓(𝑥 + 2𝜋) = 𝑓(𝑥). 07

OR

Q.3 (a) Find the Fourier series of 𝑓(𝑥) = 𝑥 + 𝜋 − 𝜋 < 𝑥 < 𝜋 𝑎𝑛𝑑 𝑓(𝑥 + 2𝜋) = 𝑓(𝑥). 07

(b) Find the Fourier series of 𝑓(𝑥) = {

2 ; 𝑖𝑓 − 2 < 𝑥 < 00 ; 𝑖𝑓 0 < 𝑥 < 2

𝑝 = 4 07

Q.4 (a) 1) Show that 𝐿{𝑠𝑖𝑛𝑎𝑡} =

𝑎

𝑆2+𝑎2 .

2) Prove that if 𝐿{𝑓(𝑡)} = 𝐹(𝑆) then 𝐿 {𝑓(𝑡)

𝑡} = ∫ 𝐹(𝑆)𝑑𝑆

∞

𝑆

03

04

(b) Evaluate 1) 𝐿{𝑡𝑠𝑖𝑛𝑎𝑡}.

2) 𝐿−1 {15

𝑆2+4𝑆+29}

03

04

OR

Q.4 (a) Solve 𝑦′′ − 𝑦 = 𝑡, 𝑦(0) = 1, 𝑦′(0) = 1 by Laplace transforms method. 07

(b) Evaluate

1) 𝐿 {1−𝑐𝑜𝑠2𝑡

𝑡}

2) 𝐿−1 {𝑒−2𝜋𝑆−𝑒−8𝜋𝑆

(𝑆2+1)}

07

Q.5 (a) 1) Form the partial differential equation from

𝑧 = 𝑓(𝑥 + 𝑦 + 𝑧, 𝑥2 + 𝑦2 + 𝑧2) = 0.

2) Define (a) Beta function (b) Heaviside’s function.

03

04

(b) Find the Fourier integral representation of the function 𝑓(𝑥) = {

1 ; 𝑖𝑓 |𝑥| < 1

0 ; 𝑖𝑓 |𝑥| > 1.

07

OR

Q.5 (a) 1) Solve 𝜕2𝑧

𝜕𝑥2 = 𝑧.

2) Form the partial differential equation from 𝑧 = 𝑥𝑦 + 𝑓(𝑥2 + 𝑦2).

03

04

(b) 1) Solve 𝑝2 + 𝑞2 = 𝑥 − 𝑦

2) Solve (𝑦 + 𝑧)𝑝 + (𝑧 + 𝑥)𝑞 = 𝑥 + 𝑦

03

04

*************

1


GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTER–III(OLD) • EXAMINATION – WINTER 2016


Subject Name:Advanced Engineering Mathematics (New)

Time: 10:30 AM to 01:30 PM Total Marks: 70 Instructions:




Q.1 (a) Attempt the following

(i) Solve dyyxydxxyx )()( 22 04

(ii) Solve xxy

dx

dy2sintan

03

(b) Find power series solution of the equation xyy 2' 07

Q.2 (a) Solve the p.d.e. yxx uu 16 07

(b) Attempt the following

(i) Solve xeyDD 22 )6( 04

(ii) Solve xyDD cos)( 3 03

OR


(i) Solve xxeyDD 42 )65( 04

(ii) Solve 12)8( 43 xxyD 03

Q.3 (a) Find Fourier series for xxxxf .)( 2 07

(b) Find Fourier series for 11,)( 2 xxxf 07

OR

Q.3 (a) If

xx

xxxf

0,1

0,1)( . Find the Fourier series for f(x).

07

(b) Find a Fourier sine series for xxxxf 0,)( 2 07

Q.4 (a)

Solve tydt

yd2cos244

2

2

given that at 3.0 yt and .4dt

dy

07

(b) Using convolution theorem find

)( 222

1

ass

aL

07

OR


(i) Find 2)4)(4( ttuL 04

(ii) Find tteL 4 03


(i) Find

3

21

s

eL

s

04

(ii) Find

52

22

1

ss

sL

03


2

(i) Define (1) Gamma Function and find

25 (2) Sawtooth wave function.

04

(ii) Form partial differential equation for ))(( byaxz 03

(b) Express the function

2,0

2,2)(

x

xxf as a Fourier Integral.

07

OR


(i) Solve the p.d.e. xxy uu 04

(ii) Find the complete integral of 1))(( qypxzqp 03


(i) Solve 222 zqypx 04

(ii) Solve 422 qp 03

*************



BE - SEMESTER–III EXAMINATION – SUMMER 2016


Subject Name:Advanced Engineering Mathematics Time:10:30 AM to 01:30 PM Total Marks: 70 Instructions:


2. Make suitable assumptions wherever necessary. 3. Figures to the right indicate full marks.

1


GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTER–III EXAMINATION – WINTER 2015



Time: 2:30pm to 5:30pm Total Marks: 70 Instructions:





(i) Solve .sintan2 xxy

dx

dy

4

(ii) Solve .0sincos)1( xdyexdxe yy 3


(i) Solve .)65( 42 xexyDD 4

(ii) Define triangular wave function and draw its graph. 3


(i) Solve .5sin40)98( 2 xyDD 4

(ii) Find ordinary and singular points for .0)2('3'')2(2 2 yxxyyxx 3


(i) Solve ).1log(cos4')1('')1( 2 xyyxyx 5

(ii) Define beta function. Find B(5,4). 2

OR

(b) Find the power series solution about x=0 of .0''' 2 yxxyy 07

Q.3 (a) Express xxxf cos)( as a Fourier series in ).,( 07

(b) Find Fourier series expansion of the function given by

20,1

02,0)(

x

xxf

07

OR

Q.3 (a) If

2,2

0,)(

xx

xxxf , find the Fourier series.

07

(b) Prove that .0,

2sin

1

2 1

lxl

xn

n

lx

l

n

07


(i) Evaluate .

3

4log1

s

sL

4

(ii) Find .4sin2 ttL 3


(i) Find the Laplace transform of the periodic function defined by

,30,2

)( tt

tf ).()3( tftf

4

2

(ii) Find Laplace inverse of .

)(

13ass

3

OR


(i) Use convolution theorem to find .

)(

122

1

assL

4

(ii) Find .

2cos1

t

tL

3


(i) Solve by Laplace transform: ,42 y

dt

dygiven that .1,0 yt

5

(ii) Find ).cos*( 2 ttL 2


(i) Derive partial differential equation by eliminating a and b from

abbyaxz .

3

(ii) Find the complete integral of .2ppqq 4

(b) Solve the p.d.e. yxx uu 16 by using separation of variables method. 07

OR


(i) Form a partial differential equation by eliminating the arbitrary function

from .0),( 222 zyxzyx

3

(ii) Solve .0252 tsr 4

(b) Using Fourier integral show that

x

xdx

,0

0,2sin

cos1

0

07

*************

1


GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTER– III EXAMINATION – SUMMER 2015

Subject Code:130002 Date:06/06/2015 Subject Name: Advanced Engineering Mathematics Time: 02.30pm-05.30pm Total Marks: 70 Instructions:


Q.1 (a) Attempt the following (i) Solve 0]1[]1[ // =−++ dy

yxedxe yxyx 04

(ii) Solve xexy

dxdy 1

2 6=+ 03

(b) Attempt the following (i) Solve xeyDD 23 )67( =+− 04 (ii) Define square wave function and draw its graph 03 Q.2 (a) Attempt the following (i) Solve xxyD 2sin2cos)9( 2 +=+ 04 (ii) Find the ordinary and singular points of 0)3(62 '''2 =+++ yxxyyx 03 (b) Attempt the following (i) Solve the Cauchy-Euler equation )sin(log53 222 xxyxDyyDx =+− 05 (ii) Define Gamma Function and obtain its value for 7. 02 OR (b) Find the series solution of 09)1( '''2 =−++ yxyyx 07 Q.3 (a)

Find Fourier series for

≤≤−≤≤

=21),2(

10,)(

xxxx

xfππ

07

(b) Attempt the following (i) Find Fourier series expansion of ππ <<−= xxxf ,)( 04

(ii) Find Fourier sine series for 2)( xxxf −= π in ).,0( π 03 OR

Q.3 (a) Attempt the following (i) Obtain Fourier series for .11,)( 2 <<−−= xxxxf 04 (ii) Find a cosine series for .0,)( π<<= xexf x 03 (b)

Obtain Fourier series to represent 2

2)(

−

=xxf π in the interval .20 π<< x

07

Q.4 (a) Attempt the following (i) Find the inverse Laplace transform of

)2()1(54

2 +−+

sss 04

(ii) Find the Laplace transform of tte t cos2sin4 03 (b) Attempt the following (i) Solve by Laplace transform .0)0(,2)0(,16 ''' ===+ yyyy 05

2

(ii) Find the convolution of 1*1 02

OR Q.4 (a) Attempt the following

(i) Find the inverse Laplace transform of )11)(6(

522 +−

−ss

s 04

(ii) Find Laplace transform of .3cosh2 tt 03


(i) Solve by Laplace transform tt eeyy 42' 24 +=− given that at .0,0 == yt 05 (ii) Find Laplace transform of ).1()1( 2 −− tut 02


(i) Derive partial differential equation by eliminating constants a and b from ).)(( byaxz ++=

03

(ii) Solve by separation of variables method: .)(2 uyxuu yx +=+ 04

(b) Use Frobenius method to solve .0)1(2 2'''2 =−+− yxxyyx 07

OR Q.5 (a) Attempt the following

(i) Form the partial differential equation by eliminating the arbitrary functions f and F from the relation ).()( atxFatxfy ++−=

03

(ii) Find the complete integral of .4zpq = 04

(b) Express the function

≥

≤=

10

11)(

xfor

xforxf as a Fourier integral. Hence evaluate

λλ

λλ dx∫∞

0

cossin .

07

*************

1


GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTER–III • EXAMINATION – WINTER • 2014

Subject Code: 130002 Date: 06-01-2015


Time: 02.30 pm - 05.30 pm Total Marks: 70 Instructions:




Q.1 (a) (1) Find the differential equation of the family of circles of radius r whose

centre lies on the x-axis.

(2) Solve

04

03

(b)) Find the series solution of y' - 2xy = 0 07

Q.2 (a) Solve by method of separation of variables

( ) 07

(b) Solve in series the differential equation 4

+ 2

+ y = 0 07

OR

(b) (1) Solve the Initial Value Problem y''- 9y = 0 ; y(0) = 2, y'(0) = -1

(2) Solve (D2 +16)y = x

4 + e

3x + cos3x

03

04

Q.3 (a) (1) Find the Laplace transform of f(t) = 0 ; 0 ≤ t ≤ 3

= 4 ; t ≥ 3

(2) Prove that L (sin h kt) =

03

04

(b) (1) Find {

( )}

(2) Find {

( )( )( )}

03

04

OR

Q.3 (a) If L{f(t)}= F(s) then show that L * ( )+ = ( )

* ( )+ ; n =1,2,3….

and use this result find L(t2sin wt)

07

(b) Solve the differential equation by Laplace Transform

y'' + y = sin2t ; y(0) = 2, y'(0) = 1 07

Q.4 (a) Find the Fourier series of f(x) = x+| x| ; - π < x < π 07

(b) Find the Fourier expansion f(x) = x2 - 2 ; -2 ≤ x ≤ 2

07

OR

Q.4 (a) Obtain the cosine series for the function f(x) = ex in the range (0,l) 07

(b) Find the Fourier series for the periodic function f(x)

f(x) = -k; if -π < x < 0 f(x + 2π) = f(x)

= k; if 0 < x < π

Hence deduce that

07

Q.5 (a) (1) Define the following terms:

(i) Beta function

(ii) Heaviside’s function

(2) Eliminate the arbitrary function from the equation

z = xy + f (x2 +y

2)

04

03

(b) Using Fourier integral representation, show that 07

2

∫

= 0 ; x < 0

= π/2 ; x = 0

= πe-x

; x > 0

OR

Q.5 (a) (1) Solve (y + z) p + (z + x) q = x + y

(2) Solve p2 +q

2 = npq

04

03 (b) (1) Solve

cos( 2x+3y)

(2) Solve ( D2 +10DD' +25D'

2) z = e

3x+2y.

04

03

*************

1


GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTER–III • EXAMINATION – WINTER 2013

Subject Code: 130002 Date: 05-12-2013 Subject Name: Advanced Engineering Mathematics Time: 02.30 pm - 05.30 pm Total Marks: 70 Instructions:


2

***************

1/3


GUJARAT TECHNOLOGICAL UNIVERSITY BE - SEMESTER–III • EXAMINATION – SUMMER 2013

Subject Code: 130002 Date: 01-06-2013 Subject Name: Advanced Engineering Mathematics Time: 02.30 pm - 05.30 pm Total Marks: 70 Instructions:


Q.1 (a) (i) Solve ( ) xy

dydxy −=+ −12 tan1 .

03

(ii) Solve 0

cossinsincos

=++++

+xyxxyyxy

dxdy .

04

(b) Find power series solution of the of the equation

02

2

=+ xydx

yd in powers of x .

07

Q.2 (a) Solve 02 =

∂∂

−∂∂

yuy

xux using method of separation of variables.

07

(b) Find the power series solution of the equation ( ) 012 =−′+′′+ xyyxyx about 0=x .

07

OR (b) Solve ( ) 16 223 +=−− xyDDD 07 Q.3 (a) Find the fourier series of the periodic function ( )xf with period π2 ,

defined as follows:

( )

<≤≤<−

=π

πxforx

xforxf

0,0,0

07

(b) Find the fourier cosine series for ( ) cxxxf ≤<= 0,2 . Also sketch ( )xf .

07

OR Q.3 (a) Find the fourier series of periodic function with period 2, which is

given below.

( )

≤≤≤≤−

=10010

xxx

xf

07

(b) Find the fourier series of the periodic function with period 2 of

( ) ( )

≤≤−≤≤

=21210

xxx

xfπ

π

07

Q.4 (a) (i) Prove that ( ) asas

eL at −>+

=− ,1 03

(ii) Prove that ( ) nsntL n

n ,!1+= being positive integer. 04

(b) (i) Find

++−

bsasL log1

04

2/3

(ii) Find ( )

++

+−22

1

542

sssL

03

OR Q.4 (a) If ( )sf is the laplace transform of ( )tf and 0≥a , then

prove that ( ) ( )[ ] ( )sfeatuatfL as−=−−

07

(b) (i) Find the laplace transform

∫ −t

x dxxeL0

cos 03

(ii) Find the laplace transform

∫ ∫t t

duduauL0 0

sin 04

Q.5 (a) (i) Define Beta function and Rectangle function. 04 (ii) Form partial differential equation ( ) ( )22 32 −+−= yxz 03 (b) Find the fourier transform of f defined by ( ) tetf −= .

Sketch the graph.

07

OR Q.5 (a) (i) Eliminate the function f from the relation ( ) 0,2 =+++ zyxzxyf . 04

(ii) Solve yx

yxz sinsin

2

=∂∂

∂ , given that yyz sin2−=

∂∂ , when 0=x , and

0=z , when y is an odd multiple of 2π .

03

(b) (i) Solve ( ) ( ) ( )yxzqxzypzyx −=−+− 04 (ii) Solve 22qpqypxz ++= 03

*************

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