Engineering Mathematics-III Important University Questions Unit-i Fourier Series Two Marks

1

ENGINEERING MATHEMATICS-III IMPORTANT UNIVERSITY QUESTIONS UNIT-I FOURIER SERIES TWO MARKS 1) Determine bn in the Fourier series expansion of f ( x ) =

0 < x < 2π with period 2π . 2) Define root mean square value of

3) If

 c x, oi 0 0 .

(Nov/Dec 2005) 1 0 < x < 1 f ( x) =  x >1 0

1

x 1

Find the Fourier transform of f ( x ) = 0

19)

Find the Fourier cosine transform of f(x) defined as  x for 0 < x 0. (May 2006) a a If F { f ( x )} = f ( s ) then give the value of F { f ( ax ) } .

18)

20) 21) 22)

. (April 2004)

d n f ( x)  and b) F   in terms of the Fourier transform of n  dx 

f(x). (Nov 2004) 11) State Fourier integral theorem. 12)

x 0

if

2 2 e −a x

for any a>0 and hence

is self-reciprocal under Fourier Cosine transform.

(May/June 2009) 2 if   2 f ( x ) = a − x , if  0, 

30) Find the Fourier transform of

deduce that 31) Find

∞ sin t − t cos t π dt = . ∫ 3 4 t 0

−ax  FC  e   

,

 1   FC    1 + x2 

Fourier Cosine transform)

and

x a>0

(May/June 2009)

 x  . FC    1 + x2 

(Hence FC stands for

(May/June 2009)

UNIT – III PARTIAL DIFFERENTIAL EQUATIONS

9

10

TWO MARKS ∂2 z = sin y . ∂x 2

1)

Solve

2)

Find the complete integral of pq = q + p + pq . (M/J 2007) (Dec 2008) Form the PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from the

3)

(May/June 2007) z

x

y

equation ( x − a ) 2 + ( y − b ) 2 = z 2 ( cot 2 α ) .(Nov/Dec 2007)(May/Jun 2009) 4)

Find the complete solution of the PDE

p 2 + q 2 − 4 pq = 0 .(Nov/Dec2007)

5)

Form the PDE of all spheres whose centers lie on the z-axis. (N/D 2006)

6)

Find complete integral of the PDE ( 1 − x ) p + ( 2 − y ) q = 3 − z .(N/D’06)

7)

obtain the PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from the equation ( x − a ) 2 + ( y − b ) 2 + z 2 = 1 .

(Nov/Dec 2003)

∂2 z ∂2 z ∂2 z − 12 + 9 = 0 .(Nov/Dec ∂x∂y ∂x 2 ∂y 2

8)

Find the general solution of 4

2003)

9)

 xy  Eliminate the arbitrary function ‘f ’ from z = f   and form the PDE.  z 

(Apr/May 2004) 10)

Find the Complete integral of p+q=pq.

(Apr/May 2004)

11)

Find the PDE of all planes passing through the origin. (N/D 2005)

12)

Find the particular integral of

13)

Find the PDE of all the planes having equal intercepts on the x & y axis.

( D − 3D D′ − 4DD′ + 1 D′ 2) Z = s ( ix + n2y) 3 2

2

3

.

(Nov/Dec 2005)

(Nov/Dec 2005) 14)

Find the solution of

px 2 + qy 2 = z 2 .

(Nov/Dec 2005)

15)

Find the PDE of all spheres having their centers on the line x=y=z . (Oct/Nov 2002)

16)

Solve

17)

Solve

( D − 3 D ′ + 2 D′ ) Z = 0

∂3 z ∂3 z ∂3 z ∂3 z − 2 − 4 + 8 =0. ∂x 3 ∂x 2 ∂y ∂x∂y 2 ∂y 3

3 2 3

.

(Oct/Nov 2002)

(Apr/May 2003)

11 18)

Form the PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from

(

)(

z = x 2 + a 2 y 2 + b2

).

(Nov/Dec 2004)

19)

Solve ( D 2 − DD ′ + D ′ − 1) Z = 0

20)

Form the PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from

(Nov 2004)

( D + 3 D D′ − D′ ) Z = 0

z = ax n + by n .

(Apr/May 2005) (Dec 2008)

3 2 3

21)

Solve

22)

Form the PDE by eliminating the arbitrary function from z=f(xy).

23)

Write down the complete solution of

24)

Form the partial differential equation by eliminating the arbitrary constants ‘a’ and ‘b’ from

z = px + qy + c 1 + p 2 + q 2

(Apr 1995) (Dec 2008)

z = ax 3 + by 3

25)

Find the singular solution of

26)

Find the general solution of px + qy = z .

27)

Find the particular integral of

28)

Solve

p + q = 1 (Dec

29)

Solve

3 2  D +D +D ′ Z =0  

30)

Form the p.d.e by eliminating a and b from

31)

Solve

32)

Give the general solution of

33)

Solve

34)

Form the partial differential equation by eliminating

p + q =x+y

z = px + qy + p 2 + q 2 + 1 .

(Dec 2008)

(Dec 2008)

2 3 x +4 y  D −4 DD ′ Z =e  

.(Dec 2008)

2008) (Dec 2008) z = a ( x + y ) + b .(Dec

2008)

(Dec 2008)

 D2 + 3DD′ + 2D′ 2  Z = 0  

2 2 +x + y z =f x + y    

.

∂2 z =0 ∂x∂y

.(Dec 2008)

(Dec 2008)

f

from the relation

.(May/June 2009)

SIX MARKS. 1)

Form the PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from the expression ( x − a ) 2 + ( y − b ) 2 + z 2 = c 2 . (May/June 2007)

(2D − 5D ′ + 2D′ )Z= 5s (2x+ iy) n 2

2

12

2)

Solve

.

3)

Solve x ( y 2 + z ) p + y ( x 2 + z )q = z ( x 2 − y 2 ) .

4)

Solve p(1+q)=qz.

5)

Solve

6)

Solve

7)

Solve z 2 ( p 2 + q 2 ) = x 2 + y 2 .

8)

Solve

9)

Find the singular integral of

10)

Solve

11)

Solve ( 3 z − 4 y ) p + ( 4 x − 2 z ) q = ( 2 y − 3 x ) .(N/D 2006), (N/D 2003)

12)

Solve

13)

Find the singular integral of PDE

14)

Solve

15)

Find the general solution of x ( z 2 − y 2 ) p + y ( x 2 − z 2 )q = z ( y 2 − x 2 ) .

16)

Solve

17)

Solve x ( y − z ) p + y ( z − x ) q = z ( x − y ) .

( D − D ′ − 2D′ ) Z = 2 x + 3 y + e

(May/June 2007)

(May/June 2007)

( x 2 − y 2 − z 2 ) p + 2 xyq = 2 zx .

2 2

3x+ 4 y

(Nov/Dec 2007) (Nov/Dec 2007)

( D − 5 D ′ + 6 D′ ) Z = y s x i n 2

(Nov/Dec 2007) (May/June 2009)

. (Nov/Dec 2007)

(D2 + 3DD′ − 4D′ 2 )z = x + s yi .n

2

(May/June 2007)

z = px + qy + p 2 + pq + q 2 .

(Nov/Dec 2006)

. (Nov/Dec 2006)

(D2 + 2DD′ + D′ 2 )z = x2 y + e x− y .

(Nov/Dec 2006) z = px + qy + p 2 − q 2 .(N/D

2003)

(D2 + 4DD′ − 5D′ 2 )z = 3e2x− y + s ( ix − n2 y) .(N/D 2003) ( D 2 − 2 DD′ + D′ 2 − 3 D + 3 D′ + 2)z = (e 3 x + 2e − 2 y ) 2 .

(N/D 2003)

(Apr/May 2004)

2 x+ y ′ ′ (D − 7DD − 6D )z = s ( xi+ 2ny) + e 3 2 3

13

18)

Solve

. (A/M 2004)

19)

Solve ( x + y ) zp + ( x − y ) zq = x 2 + y 2 .

20)

Solve

21)

Solve

(Nov/Dec 2005)

( D 2 + 2 DD ′ + D′ 2 + 2 D + 2 D′ + 1)z = e 2 x + y . z = px + qy + 1 + p 2 + q 2

(N/D 2005)

.

(May/June 2009) (Nov/Dec 2005),(Apr/May 2004) ∂2 z ∂2 z ∂2 z + − 6 = y cos x . ∂x∂y ∂x 2 ∂y 2

22)

Solve

23)

Solve ( x 2 + y 2 + yz ) p + ( x 2 + y 2 − xz )q = z ( x + y ) .(N/D 2005)

24)

Solve

( D 2 − DD ′ − 2 0D ′ 2 ) z = e 5 x + y + s in( 4 x − y ) .(Nov/Dec 2005)

25)

Solve

z = p2 + q2 .

26)

Solve

(D2 + DD′ − 6D′ 2 )z = x2 y + e3x+ y .

27)

Form the PDE by eliminating the arbitrary functions f and g in

(

)

(

z = f x3 + 2y + g x3 − 2y

(Nov/Dec 2005)

(Nov/Dec 2005) (Nov/Dec 2005)

).

(Oct/Nov 2002)

28)

Solve ( y − xz ) p + ( yz − x ) q = ( x + y )( x − y ) . (Oct/Nov 2002)

29)

Solve

( D 2 − DD ′ − 3 0D ′ 2 )z = x y+ e 6 x + y .

30)

Solve

z 2 = 1 + p2 + q2 .

31)

Solve ( y − z ) p − ( 2 x + y ) q = 2 x + z .

32)

Form the PDE by eliminating the arbitrary functions f and g in

(April 1996)

(Dec 2008)

(Apr/May 2003)

z = x 2 f ( y ) + y 2 g( x )

33)

Form the p.d.e by eliminating the function f and g from z = f ( x + 2 y ) + xg ( x + 2 y )

(Dec 2008)

34)

Solve yp + xq = z

35)

Solve

36)

Solve

37)

Form the p.d.e by eliminating the arbitrary function f and g from

(Dec 2008) (Dec 2008)

p2 x2 +q2 y2 = z 2 3 2  D −2 D D ′ z =sin  

z = f ( x + 2 y ) + xg ( x + 2 y )

( x +2 y ) +3 x 2 y

(Dec 2008) (Dec 2008)

14 38)

Solve

2 +z 2 p −xyq +xz =0  y   

39)

Solve

r + s − 6 t = y cos x

40)

Obtain complete solution of the equation

41)

Solve

42)

Solve xzp + yzq = xy

43)

Solve

44)

Find the complete solution of

45)

Solve the equation

 D2 + DD′ − 6D′ 2  z = c ( x2 o+ y) s   .

(Dec 2008) (Dec 2008) z = px + qy − 2

pq

(Dec 2008)

(Dec 2008)

(Dec 2008)

 2D2 − 5DD′ + 2D′ 2  z = e2x+ y  

(Dec 2008)

pqxy

(May/June 2009)

=z2

 D2 − D′ 2  Z = ex− y s ( xi+ 2ny)  

.

(May/June 2009)

UNIT-IV APPLICATIONS OF PDE

TWO MARKS 1) Classify the following second order partial differential equations: i) 4

∂2 u ∂2 u ∂2 u ∂u ∂u + 4 + −6 −8 − 16 u = 0 . ∂x∂y ∂y 2 ∂x ∂y ∂x 2 2

(Apr/May 2003)

2

 ∂u  ∂ 2 u ∂ 2 u  ∂u  ii) 2 + 2 =   +   . (Apr/May 2003) ∂x ∂y  ∂x   ∂y  2 2 iii) y u xx − 2 xyu xy + x u yy + 2u x − 3u = 0 . (Nov/Dec

2003) iv) y u xx + u yy + + + 7 = 0 . (Nov/Dec 2003) u + xu = 0 xx yy v) . (Apr/May 2004) 2) Classify the partial differential equations 2

3)

∂2u 1 ∂u = 2 . 2 ∂x α ∂t

u x2

u 2y

(May/June 2007)

Classify the following PDE i) (1 + x ) 2 u xx − 4 xu xy + u yy = x . ii) x 2 u xx + 2 xyu xy + (1 + y ) 2 u yy − 2u x

(March 1998), (Apr/May 2004) (Dec 1998)

=0.

15 2

4)

What is the constant a in the wave equation utt = a u xx ? (N/D 2004)

5)

2 ∂u 2 ∂ u = α In the diffusion equation what does α2 stand for? 2 ∂t ∂x

6)

2

(N/D 2005) (Dec 2008) What is the basic difference between the solutions of one dimensional wave equation and one dimensional heat equation? (Nov/Dec 2005)

7)

What are the possible solutions of one dimensional wave equation? (May/June 2006)

8)

Explain the various variables involved in one dimensional wave equation. (April 1995),(Nov 1995)

9)

A tightly stretched string of length 2L is fastened at both ends. The midpoint of the string is displaced to a distance ‘b’ and released from rest in this position. Write the initial conditions.

10)

(May 2006)

Write the initial conditions of the wave equation if the string has an initial displacement but no initial velocity. (A.U.Tri. Nov/Dec 2008)

11)

Write the boundary conditions and initial conditions for solving the vibration of string equation , if the string is subjected to initial displacement f(x) and initial velocity g(x).

12)

(Nov/Dec 2006),(April 1998)

State one dimensional heat equation with initial and boundary conditions. (Nov/Dec 2006)

13)

In steady state conditions derive the solution of one dimensional heat flow equation.

14)

(Nov/Dec 2005)

An insulated rod of length 60 cm has is ends A and B maintained at 20  C and 80  C respectively. Find the steady state solution of the rod. (Nov/Dec 2003)

15)

A rod 30 cm long has its ends A and B kept at 20  C and 80  C respectively until steady state conditions prevail. Find the steady state temperature in the rod.

16)

17)

(Apr/May 2004)

State any two laws which are assumed to derive one dimensional heat equation.

(Nov/Dec 2004)

State Fourier law of heat conduction.

(Apr/May 2005)

18) What are the possible solutions of one dimensional heat equation? (May 2000)

(May/June 2009)

16 19)

How many boundary conditions are required to solve completely ∂u ∂2u =α2 ∂t ∂x 2

(April 1995)

20)

Define temperature gradient.

21)

State the assumptions made in the derivation of one dimensional wave equation.

22)

(April 1995), (Nov 1995),(Nov 2007) (Dec 2008)

Write the steady state heat flow equation in two dimension in Cartesian & Polar form.

23)

(Nov 1995)

(Nov/Dec 2005)

Write any two solutions of the Laplace equation obtained by the method of separation of variables.

(April 2003)

24) In two dimensional heat flow, the temperature at any point is independent of which coordinate? 25) Explain the term steady state. 2  4 + x 2 u 2 u  1 + x    xx +  5 + 2 x   xy + u yy = 0     

26)

Classify the p.d.e

(Dec 2008)

27)

State the empirical laws used in deriving one-dimensional heat flow equation. (Dec 2008) r 2 urr + ru r + uθ θ = 0 .

28)

Write the product solutions of

29)

What is the equation governing the two dimensional heat flow steady state and also write its solution.

(Dec 2008) (Dec 2008)

∂2 u ∂2 u ∂2 u +2 + = e 2 x +3 y . ∂x∂y ∂x 2 ∂y 2

30)

Classify the p.d.e

31)

Write the various possible solutions of the Laplace equation in two dimensions.

32)

(Dec 2008)

A infinitely long uniform plate is bounded by the edges x = 0 , x = l and the ends right angles to them. The breadth of the edges maintained at

f ( x) .

All the other edges are kept at

boundary condition in mathematical form. 33)

y =0 0 C .

is l and is Write down the

(Dec 2008)

Write any two assumptions made while deriving the partial differential equation of transverse vibrations of a string.

34)

(Dec 2008)

(Dec 2008)

Define steady state. Write the one dimensional heat equation in steady state.

(Dec 2008)

17 35)

Write all the solutions of Laplace equation in Cartesian form, using the method of separation of variables.

36)

Verify that

y = cosh ( λx ) cosh ( − λat )

(Dec 2008)

is a solution of

2 ∂2 y 2 ∂ y = a . (M/J’09) ∂t 2 ∂x 2

12 MARKS 1)

A tightly stretched string of length ‘ l ’ has its ends fastened at x=0 & x=l . The midpoint of the string is then taken to a height ‘ h ’ and then released from rest in that position. Obtain an expression for the displacement of the string at any subsequent time.

2)

(Nov 2002)

A tightly stretched flexible string has its ends fixed at x=0 and x=l. At time t=0 , the string is given a shape defined by

f ( x ) = kx 2 ( l − x ), where

k is a

constant , and then released from rest. Find the displacement of any point x of the string at any time 3)

t > 0.

(April 2003)

A tightly stretched string with fixed end points x=0 and x=l is initially in a πx   . It is released from rest from this  l 

3 position given by y( x ,0) = y0 sin 

position. Find the displacement at anytime ‘ t ’. 4)

(Nov 2004)

A tightly stretched string of length ‘ 2l ’ has its ends fastened at x=0 , x=2l. The midpoint of the string is then taken to height ‘ b ’ and then released from rest in that position. Find the lateral displacement of a point of the string at time ‘ t ’ from yhe instant of release.

5)

(May 2005)

A string of length ‘ l ’ has its ends x=0 , x=l fixed. The point where x =

l is 3

drawn aside a small distance ‘ h ’,the displacement y ( x , t ) satisfies 2 ∂2 y 2 ∂ y = a . Find y ( x , t ) at any time ‘ t ’. ∂t 2 ∂x 2

6)

An elastic string of length ‘ 2l ’ fixed at both ends is disturbed from its equilibrium position by imparting to each point an initial velocity of magnitude

7)

k ( 2lx − x 2 ). Find

the displacement function y ( x, t ) . (May ‘06)

A uniform string is stretched and fastened to two points ‘ l ’ apart. Motion is started by displacing the string into the form of the curve y = kx ( l − x ), and then releasing it from this position at time t=0. Find the displacement

18

of the point of the string at a distance ‘ x ’ from one end at time ‘ t ’. (A.U.Tri. Nov/Dec 2008) (Dec 2008) (May/June 2009) 8)

If a string of length ‘ l ’ is initially at rest in its equilibrium position and each

of its points is given a velocity ‘ v ’ such that

 v = c( l − x ) cx

l 0 < x < for 2 for l