Second Order Partial Derivatives - Engineering Mathematics - Past Paper, Exams for Engineering Mathematics. Jaypee University of Engineering & Technology

Engineering Mathematics

Description: Main points of this exam paper are: Second Order Partial Derivatives, Taylor Series Expansion, Cartesian Coordinates, Arbitrary Function, Maximum Values, Inverse Laplace Transform, Laplace Transformations, Differential Equations
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CORK INSTITUTE OF TECHNOLOGY

CORK INSTITUTE OF TECHNOLOGY

INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ

Semester 2 Examinations 2009/10

Module Title: Engineering Mathematics 211

Module Code: MATH7006

School: Building & Civil Engineering

Programme Title: Bachelor of Engineering (Honours) in Civil Engineering-Stage 2

Programme Code: CSTRU-8-Y2

External Examiner(s): Dr.P.Robinson

Internal Examiner(s): Mr. T. O Leary

Instructions: Select any four questions. The questions carry equal marks.

Duration: 2 Hours

Sitting: Autumn 2010

Requirements for this examination:

Note to Candidates: Please check the Programme Title and the Module Title to ensure that you have received the correct

examination paper.

If in doubt please contact an Invigilator.

1. (a) (i) Find a Taylor series expansion of the function

V=f(x,y)=ln(x 2 -4y)

about the values x=3, y=2. The series is to contain terms deduced from the first and

the second order partial derivatives of f(x,y).

(ii) Bt using differentials (partial derivatives) estimate the value of V where the values

of x and y were estimated to be 4±0.01 and 2±0.02, respectively. (13 marks)

(b) Write down the relationships between Cartesian coordinates x and y and polar

coordinates r and . If stress T=f(x,y) is an arbitrary function in x and y find the

relationships between the partial derivatives of T with respect to x and y and those

with respect to r and . Also show that

222

2

2

y

T

x

T

θ

T

r

1

r

T  

  

 

  

 

  

 

  

 (5 marks)

(c) Find the maximum/minimum values of V=4x 2 -8xy+y

3 +4y+8 (7 marks)

2. (a) By using partial fractions and by completing the square find the Inverse Laplace

Transform of the expression

5s6s

84s 2 

 (11 marks)

(b) By using Laplace Transformations solve the differential equations

(i) 0(0)yy(0)6ey2 dt

dy 3

dt

yd t 2

2



(ii) 0(0)yy(0)60sin2ty2 dt

dy 3

dt

yd 2

2

 (14 marks)

3. In answering the following you are required to use the Method of Undetermined

Coefficients. No marks will be awarded if any other method is used.

Select any three parts of the following:

(a) In the theory Beam Struts the differential equation below arises

2EI

Wx - yω

dx

yd 2 2

2

 where EI

P ω2 

Solve this differential equation where y=0 at x=0 and at x=L to show that

 

  

 

L

x

ωLsin

ωxsin

2P

WL y

Show that the strut fails if the load P reaches the critical value 2

2 2

L

EI ω

  .

(9 marks)

(b) Solve the differential equation

0(0)yy(0)244y dx

dy 4

dx

yd 2

2

 (8 marks)

(c) Find the general solution of the differential equation

2tsin20 dt

dy 2

dt

yd 2

2

 (8 marks)

(d) Find the general solution of the differential equation

t

2

2

6e2y dt

dy 3

dt

yd  (9 marks)

4. (a) R is the triangular region with vertices (-1,0), (1,0) and (1,4).

(i) If C is the perimeter of this region evaluate the line integral

  C

22 dy12ydx6x

(ii) Evaluate the double integrals

 R

3dA80y (13 marks)

(b) A volume V is in the form of a right circular cylinder and is described by

V: x 2 +y

2 ≤4 0≤z≤3

(i) Evaluate the line integral

  C

2 4xydydx3x

where C is the perimeter of the base.

(ii) For this volume V evaluate the triple integral

 V

2zdV4x

Note: cos 2 A=

1

2 (1+cos2A) sin

2 A=

1

2 (1-cos2A) (12 marks)

5 In this question there is a choice in part (c).

(a) (i) Consider the set of simultaneous differential equations

0y(0)y3x dt

dy

6x(0)yx dt

dx





By forming a second order differential equation find the general solution for x and for y

or

By using Laplace Transforms solve for x. (7 marks)

(b) The differential equation below can be solved by separating the variable.

5y(0)2xyy dx

dy 

(i) Solve this differential equation.

(ii) By using Eulers Method with a step of 0.1 estimate the value of y at x=0.2.

Calculate the error in this approximation. (9 marks)

(c) By using the Method of Variation of Parameters find the general solution of the

differential equation

x 2

2

16xey dt

yd 

Note:   2 axax

ax

a

e

a

xe dxxe where “a” is a constant. (9 marks)

or

(c) Solve the differential equation

0(0)yy(0)8e1y10 dx

dy 2

dx

yd x 2

2

 (9 marks)

DERIVATIVES

f(x) a=constant (x)f  nx 1nnx 

lnx

x

1

axe a axe

sinx cosx

cosx -sinx

uv

dx

du v

dx

dv u 

tan -1  

  

a

x 2 2

a

x +a

tan -1

x 2

1

x +1

v

u

2v

dx

dv u

dx

du v 

INTEGRALS

f(x) a=constant f(x)dx nx n+1x

if n -1 n+1

x

1

lnx

axe 1

a a axe

sinx -cosx

cosx sinx

LAPLACE TRANSFORMS

For a function f(t) the Laplace Transform of f(t) is a function in s defined by

F(s) e f(t)dt st

0

  

 where s>0.

f(t) F(s)

A=constant A

s

t n

1ns

n! 

eat 1

s a

sinhkt k

s k2 2

coshkt s

s k2 2

sin t 

s2 2

cos t s

s2 2

e f(t)at F(s-a)

f (t) sF(s)-f(0)

f (t) s F(s) sf(0) f (o)2   

Note: coshA e e

2 sinhA

e e

2

A A A A

 

  

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