s e sem iii maths tut

8/11/2019 s e Sem III Maths Tut

http://slidepdf.com/reader/full/s-e-sem-iii-maths-tut 1/5

KONKAN GYANPEETH COLLEGE OF ENGINEERING,KARJAT.

TUTORIAL NO 1

Class: - S.E. Subject: - Applied Maths- III

Sem: - III Academic Year- 2014-15

Q. 1 Find Laplace transform of the following functions.

1) sinh5

t

2)

2sin

t e

t t

3) ( )22sinh t t

4)t

t t t

e cosh2sin2−

5)2sin

t

t

6) t erf t t

e 43

7) −−

t duu

ueu

0

sin1

8) −

t duu

uue

0

22cos3

Q. 2 Evaluate the following.

1) ∞ −

0

2sin

t

dt t t

e

2) dt t erf t t

e 98

0

−∞

Q. 3 Find the Laplace transform of f(t) = |sin pt|, t >= 0 (Periodic)




TUTORIAL NO 2



Q. 1 Find Inverse Laplace Transform of following functions.

1)( )( )3

13

2

++

+

ss

s 2)

( )( )2252

42

22

2

++++

−+

ssss

ss

3)( )44

4as

s

+ 4)

−

2

1 2tan

s

Q. 2 Find Inverse Laplace Transform of following using Convolution theorem.

1)

( )( )2252

3222

2

++++

++

ssss

ss 2)

+−

b

as

s

1tan

1

Q. 3 Express the following functions in terms of Heaviside’s unit step function and hence

find their Laplace transform

( )

>

≤<

≤<

=

π

π π

π

2,3sin

2,2sin

0,sin

t t

t t

t t

t f

Q. 4 Find the Laplace transform of ( )π −− t H t e t sin

Q. 5 Evaluate using Laplace transform ( )dt t H t t e t

∞

−

−++0

2 2)31(

Q. 6 Find inverse Laplace transform of ( )258

2

2

++

−

ss

e s

Q. 7 Solve following differential equations by using Laplace Transform.

t ydt ydt

dy t

sin20

=++ given that y(0) = 1




TUTORIAL NO 3



1) Find k such that ( ) y

kxi y x

122tanlog

2

1 −++ is analytic.

2) Show that f(z) = cos z is analytic and find its derivative.

3) If u and v are harmonic conjugate functions, show that uv is a harmonic function.

4) Find the orthogonal trajectory of the family of curves c x y x =+−22

5) Find the analytic function whose imaginary part is )cossin( y x y ye x+

−

6) Find the analytic function such that y

e ye x

ye x x

vu−

−−

−−+

=−

cos2

sincos when 0

2=

π f

7) Find an analytic function where, )cos(sin y yevu x+=+

8) Find image of triangle whose vertices are (0,1),(2,2),(2,0), under the transformation w = 4z

9) Find bilinear transformation under which i, -1, 1of z-plane are mapped to 0, 1, ∞ of w-

plane.

10) Find the bilinear transformation which maps points z = 1, i, -1 onto points w = i, 0, -i.

Hence find fixed points of the transformation and image of | z | < 1




TUTORIAL NO 4



1) Prove that the points (2,1,1), (0,1,-3), (3,2,-1), (7,2,7) are coplanar.

2) Prove that, ( ) ( ) ( ) ak ak ja jiai 2=××+××+××

3) Find the values of a, b, c if the directional derivative of )1,2,1(322

−++= at xczbyzaxyφ

has maximum magnitude 64 in the direction parallel to the z-axis.

4) Prove that r r r

r 3

2. −=

∇∇

5) Prove that k zx y j z y xi y x zF )2()23()32(2

+++++++= is irrotational and find scalar

potential function such that 4)0,1,1(, =ΦΦ∇=F Hence find the work done by in

moving a particle from A(0,1,1) to B(3,0,2)6) Using Green's Theorem Evaluate j y xi y xF for r d F

C

)()(. 22++−= , and C is the triangle

with vertices (0,0), (1,1), (2,1)

7) Use Stroke's theorem to evaluate xk zj yiF for r d F C

++= . and C is the boundary of the

surface 0,122

>−=+ z z y x

8) Using Gauss's Divergence theorem evaluate ++

S

dS czbyax )(222

over the sphere

1222=++ z y x




TUTORIAL NO 5



Obtain the Fourier series of following functions in the given interval

1)4

3

21

1),2,0(sin)(

2 =

∞

= −

= n

nthat Deducein x x x f π

2) .....7

1

5

1

3

1

1

1

96,

0,2

0,2)(

4444

4

++++=

<<−

<<−+

= π

π π

π π

that Deduce

x x

x x x f

3) ( )

<<−

<<=

21,2

10,

)( x x

x x

x f π

π

4) Find half range sine series for

..7

1

5

1

3

1

1

1

32),,0()(

3333

32

+−+−=−= π

that Deducelin xlx x f ,

5) Find half range cosine series for90

1

1),,0()()(

4

4

π π π =

∞

=

−= n

nthat Deducein x x x f

6) Find complex form of Fourier series for )5,5(2sinh2cosh)( −+= in x x x f .

7) Find the Fourier transform of

>

≤−=

1,0

1,)1()(

2

x

x x x f and hence evaluate

dx x

x

x x x

2cos.

sincos

0

3∞

−

8)If f(k) = 4k U(k) and g(k) = 5

k U(k), find the Z-transform of f(k)*g(k)

s e sem iii maths tut

Documents