DEPARTMENT OF MATHEMATICS
181202- MATHEMATICS – II
PART A (Q & A), PART B QUESTION BANK
UNIT-I ORDINARY DIFFERENTIAL EQUATIONS1. / Solve
A.E is m2 – 6m + 13= 0
Hence the solution is y = e3x ( A Cos2x + B Sin2x)
2. / Solve (D2 – 2D + 1) y = ex
A.E is m2 – 2m + 1 =0
(m – 1)2 = 0 => m = 1,1
C.F => (A + Bx) ex
P.I =
Solution is y = C.F + P.I = (A + Bx) ex +
3. / Find the particular integral of ( D2 + 5D + 6) y = 11 e5x
P.I =
Put D = 5 hence P.I =
4 / Solve (D – 1) 2 y = sinh2x.
A.E is (m – 1)2 = 0
m = 1,1
C.F = (A + Bx) ex
P.I =
= >
Solution is y = C.F + P.I => (A + Bx) ex +
5. / Solve
(x2 D2 + 4xD + 2) y = 0
Put x = ez ,z = logx, xD = ,x2 D2 = ( - 1)
( ( - 1+ 4 + 2) z = 0 => (2 + 3 + 2) z = 0
A.E is m2 + 3m + 2 = 0 => m = -2, -1.
Z = A e-2z + B e-z
=> y = A e-2(logx) + B e-(logx) => y =
6. / Find the particular integral of ( D2 + 2) y = x2
P.I =
=
P.I =
7. / Find the particular integral of (D2 – 2 D + 1)y = coshx
P.I =
8. / Solve
Dy – x = 0……….(1)
y – Dx = 0………..(2)
Eliminate x from these two eqns => (D2 – 1)x = 0
A.E is m2 – 1 = 0 => m = 1, -1.
=> x = Aet + B e-t
=> = y
Hence x = Aet + B e-t , y = Aet - B e-t
9. / Solve
(D2 - 4D + 3) y = 0
A.E is m2 – 4m + 3 = 0 => m= 3, m = 1.
C.F = Ae3x + Bex
Sin3x cos2x =
P.I =
Put D2 = -52
P.I =
y = Ae3x + Bex +
10. / Transform the equation ((2x + 3)2 D2 – (2x + 3) D – 12) y = 6x into linear differential equation with constant coefficients.
Put 2x + 3 = ez , z = log ( 2x + 3)
( 2x + 3)2 D2 = 4 (2 - )
( 2x + 3)D = 2
Hence the D.E is (42 - 6 -12) y = z ez - 9
11. / Find the Particular integral of (D – 3)2 y = e3x cosx
P.I = = = -e3x cosx
12. / Find the Particular integral of (D2 + 1)y = sin2 x
P.I =
P.I =
13. / Find the Particular integral of (D2 + 4D + 4)y = e-2x x
P.I = =
14. / Transform the equationinto linear differential equation with constant coefficients.
Put x = ez ,z = logx, xD = ,x2 D2 = (-1)
the given eqn => ((-1) +6 + 2)y = ez z
(2 + 5 + 2) z = z ez
15. / Write Cauchy’s homogeneous linear equation.
where ai’s are constants and X is a function of x.
16. / Write Legendre’s linear equation.
A1 , A2,…….are constants.
17. / Find the Particular integral of (D + 2) (D – 1)2 = e-2x
P.I = =
18. / Transform the equation into linear differential equation with constant coefficients.
Put x = ez ,z = logx, xD = ,x2 D2 = (-1)
(2 - 2 + 1) z = (z e-z)2
19. / Find the Particular integral of (D2 + 4) y = sin2x
P.I =
Put D2 = -22 , P.I =
20. / Transform the equation ((3x + 5)2 D2 – 6(3x + 5) D + 8) y = 0 into linear differential equation with constant coefficients.
Put 3x + 5 = ez , z = log ( 3x + 5)
( 3x + 5)2 D2 = 9 (2 - )
( 3x + 5)D = 3
Hence the given eqn becomes ( 9 (2 - ) - 18 + 8 )z = 0 => (92 -27 + 8)z = 0.
PART B
1. / Solve ( D2 + 16 ) y = cos3x
2 / Solve by the method of variation of parameters .
3. / Solve (x2 D2 – 3 x D + 4 ) y = x2 cos(logx)
4. / Solve given x = 1 and y = 0 at t = 0.
5. / Solve (x2 D2 – 2 x D - 4 ) y = x2 + 2logx.
6. / Solve .
7. / Solve ( D2 + 4D+3 ) y = e–x sinx.
8. / Solve ( D2 + 1 ) y = x sinx by the method of variation of parameters.
9. / Solve (3x + 2)2 y’’ + 3 (3x + 2) y’ – 36 y = 3x2 + 4x + 1.
10. / Solve (D2 -1) y = x ex sinx.
UNIT-II VECTOR CALCULUS
1. / If , find grad φ at (1, -1, 2)
2. / Find at (2, -2, -1).
3. / Find the directional derivative of = 3x2+2y-3z at (1, 1, 1) in the direction.
=
4 / Find the unit normal vector to the surface x2 + xy + z2 = 4 at the point (1,-1,2).
= x2 + xy + z2 , = =
.
5. / Find the angle between the surfaces x logz = y2 – 1 and x2y = 2 –z at the point (1, 1, 1).
Let 1 = y2 –xlogz -1
1 = - log z , (1) (1,1,1) = and |1| =
Let 2 = x2y – 2+z
2 = , (2)(1,1,1) = and |2| =
,
=
6. / In what direction from (-1,1,2) is the directional derivative of = xy2z3 a maximum. Find also the magnitude of this maximum.
Given = xy2z3
and at (1,1,2) =
The maximum directional derivative occurs in the direction of
=
The magnitude of this max. directional derivative =
7. / Prove that
= = = = .
8. / If , find div curl
div (curl ) = = 0
9. / Find ‘a’, such that (3x-2y +z) is solenoidal.
Div = (3x-2y +z ) =3+a+2 = 5 + a
Div = 0 a = –5.
10. / If and are irrotational vectors prove that is solenoidal.
is irrotational curl = 0 and is irrotational curl = 0
( curl ) - ( curl ) = 0 - 0 = 0.
is solenoidal.
11. / Show that the vector is irrotational.
12. / If , evaluate where C is the part of the curve y = x3
between x =1 and x=2.
y = x3 dy = 3x2 dx
== = 31 + 63 = 94.
13. / If =, evaluate the line integral from (0,0) to (1,1) along the path y=x.
= =
14. / If .Check whether the integral is independent of the path c.
This integral is independent of the path of integration if
.Hence the line integral is independent of path.
15. / State Stoke’s theorem.
If S is an open surface bounded by a simple closed curve C and if a vector function is continuousand has continuous partial derivatives in S and on C, then = whereisthe unit vector normal to the surface.
16. / State Green’s Theorem
If M(x ,y) and N(x ,y) are continuous function with continuous partial derivatives in a region R of the xy plane bounded by a simple closed curve C, then .Where C is the curve described in positive direction
17. / State Gauss Divergence Theorem
If V is the volume bounded by a closed surface S and if a vector function is continuous partial derivative in V and on S , then
18. / Using Divergence theorem, evaluate over the surface of the sphere x2 + y2 + z2 = a2.
By Divergence theorem,
====.
19. / Find the area of a circle of radius ‘a’ using Green’s theorem.
Area = on x2+ y2 =a2 , We have x=acos y=asin , :02
Therefore Area = = = a2.
20. / If S is any closed surface enclosing a volume V and, prove that.
Div = a+b+c ; = (a+b+c) = (a+b+c)V.
PART B
1. / Verify stoke’s theorem for in the square region in the XY plane bounded by the lines x = 0, y = 0, x = a and y = a.
2. / Find the directional derivative of Ф = 2xy + z2 at the point (1,-1,3) in the direction of
3. / Using Gauss – Divergence theorem, evaluate where and
S is the surface bounded by the cube x = 0, x = 1, y = 0, y = 1, z = 0, z = 1.
4. / Verify Green’s theorem for where C is the boundary of the square formed bythe lines x = 0, y = 0, a = a, y = a.
5. / Show that is Irrotational and find its scalar potential
6. / Find the work done when a force moves a particle in thexy –
Plane from (0,0) to (1,1) along the parabola y2 = x.
7. / Find the angle between the normals to the surface xy3z2 =4 at the points (-1,-1,2) and (4,1,-1)
8. / Ir is the position vector of the point (x,y,z) , Prove that
9. / Evaluate where C is the square bounded by the line x=0 ,x=1 , y =0 and y=1.
10 / Prove that
UNIT-III ANALYTIC FUNCTIONS
1. / Define an analytic function (or) holomorphic function (or) Regular function.
A function is said to be analytic at a point if its derivative exists not only at that point but also in some neighborhood of that point.
2. / Define an entire or an integral function.
A function which is analytic everywhere in the finite plane is called an entire function. An entire function is analytic everywhere except at z =
Ex. ez, sinz, cosz, sinhz, coshz
3. / State the necessary condition for f(z) to be analytic [Cauchy – Riemann Equations]
The necessary conditions for a complex function f(z) = u(x,y) + iv(x,y) to be analytic in a region R are
(i.e) ux = vy and vx = - uy
4 / State the sufficient conditions for f(z) to be analytic
If the partial derivatives ux, uy, vx and vy are all continuous in D and ux = vy and uy= - vx. Then the function f(z) is analytic in a domain D.
5. / State the polar form of the C – R equation.
In Cartesian co – ordinates any point z is z = x + iy
In polar co – ordinates it is z = rei where r is the modulusand is the argument. Then the C- R equation in polar co – ordinates is given by
6. / State the basic difference between the limit of a function of a real variable and that of complex variable.
In the real variable, x →x0 implies that x approaches x0 along the X – axis (or) a line parallel to the X – axis.
In complex variable, z →z0 implies that z approaches z0 along any path joining the points z and z0 that lie in the Z – plane.
7. / Define harmonic function.
A real function of two real variables x and y that possesses continuous second order partial derivatives and that satisfies Laplace equation is called a harmonic function.
8. / Define conjugate harmonic function.
If u and v are harmonic functions such that u + iv is analytic, then each is called the conjugate harmonic function of the other.
9. / Define conformal transformation.
Consider the transformation w = f(z), where f(z) is a single valued function of z, a point z0 and any two curves C1 and C2 passing through z0 in the Z plane,will be mapped onto a point w0 and two curves and in the W plane. If the angle between C1 and C2 at z0 is the same as the angle between and at w0 both in magnitude and direction, then the transformation w = f(z) is said to be conformed at the point z0.
10. / Define Isogonal transformation.
A transformation under which angles between every pair of curves through a point are preserved in magnitude but opposite in direction is said to be isogonal at that point.
11. / Define Bilinear transformation.
The transformation ’ ad – bc where a, b,c,d are complex numbers is called a bilinear transformation. This is also called as Mobius or linear fractional transformation.
12. / Define Cross Ratio.
Given four points in this order , the ratio is called the cross ratio of the four points.
13. / Show that f(z) = is differentiable at z = 0 but not analytic at z = 0.
Let z = x + iy and
So the CR equations ux = vy and uy= - vx are not satisfied everywhere except at z = 0. So f(z) may be differentiable only at z = 0. Now ux = 2y, vy = 0 and uy = 2y, vx = 0 are continuous everywhere and in particular at (0, 0). So f(z) is differentiable at z = 0 only and not analytic there.
14. / Determine whether the function 2xy + i(x2 – y2) is analytic or not.
Let f(z) = 2xy + i(x2 – y2)
u = 2xy ; v = x2 – y2
ux = 2y, vy = -2y and uy = 2x , vx = 2x
ux vy and uy - vx
CR equations are not satisfied.
Hence f(z) is not an analytic function
15. / Prove that an analytic function whose real part is constant must itself be a constant.
Let f(z) = u + iv
Given u = constant. ux = 0 and uy = 0
By CR equations
ux = 0 , vy = 0 ; uy = 0 vx = 0
f1(z) = ux + ivx = 0 + i0
f1(z) = 0 f(z) = c
f(z) is a constant.
16. / Show that the function u = 2x – x3 + 3xy2 is harmonic
Given u = 2x – x3 + 3xy2
ux = 2 – 3x2 + 3y2 and ;uy = 6xy and
+ = -6x + 6x = 0
Hence u is harmonic
17. / Find a function w such that w = u + iv is analytic, if u = exsiny
Given u = exsiny
By Milne Thomson’s method
18. / Find the image of the hyperbola x2 – y2 = 10 under the transformation w = z2
w = z2
u = x2 – y2 and v = 2xy ; x2 – y2 = 10 (i.e) u = 10
Hence the image of the hyperbola x2 – y2 = 10 under the transformation w = z2 is u = 10 which is a straight line in w plane.
19. / Obtain the invariant points of the transformation
,The invariant points are given by
20. / Define a critical point of the bilinear transformation.
The point at which the mapping w = f(z) is not conformal, (i.e) f1(z) = 0 is called a critical point of the mapping.
PART B
1. / Find the fixed points of the transformation
2 / Give an example such that u and v are harmonic but u+iv is not analytic
3. / Prove that ex cos y is harmonic function.
4. / Define bilinear transformation and what the condition for this to be conformal is.
5. / If u + iu is analytic, then prove that v –iu is analytic.
6. / Find the image of the circle |z|=2 under the transformation w=3z.
7. / Determine the region of the w-plane into which the first quadrant of z-plane mapped by
the transformation w=z2.
8. / Construct the analytic function f(z) = u+iv given that 2u+3v = ex(cos y – sin y).
9. / Find the bilinear transformation that maps z = (1, i, –1) into w=(2, i, –2).
10. / Show that ex( x cos y – y sin y) is harmonic function. Find the analytic function f(z) for which ex (x cos y – y sin y) is the imaginary part.
UNIT-IV COMPLEX INTEGRATION
1. / What is the value of where C is ?
if f (z) is analytic inside and on C.Since ez is analytic on and inside , the given integral is zero.
2. / Evaluate where C is .
Here f(z) =cos, C is ,
By Cauchy’s integral Formula =2.
3. / Evaluate where C is using Cauchy’s integral formula.
Z= -2 is out of the given circle ,
4. / Evaluatewhere C is .
We know that Cauchy’s integral formula is
Given =Here f (z) = 3z2+7z+1, a=–1 lies outside. Therefore by Cauchy’s integral formula =0.
5. / Obtain the Laurent expansion of the function in the neighborhood of its
Singular point. Hence find the residue at that point.
Z=1 is a pole of order 2
Put z-1=u .Then becomes
Residue of f(z) = coefficient of =e
6. / Expand f(z)=in the region . Using this result, expand and tan-1z in
powers of z.
By using binomial series expansions,
=…….(1)
Now
Changing z by –z2 in(1) -----(2)
If f(z)=tan-1z, then f’(z)=
Hence By integrating (2) w.r.to z ,tan-1z=
7. / Obtain the Taylor’s series expansion of log(1+z) when
Let f (z) =log(1+z) f (0)=log1=0
=
8. / What is the Nature of the singularity at z=0 of the function.
f(z) = The function f(z) is not defined at z=0.But by L’Hospital’s rule,
Since the limit exists and is finite, the singularity at z=0 is a removable singularity.
9. / Evaluate where C is
Given both lies outside c.
Therefore
10. / Find the residue of the function f(z)= at a simple pole.
, z = 2 is a simple pole. Res[f(z)]z=2=
11. / State Laurent’s series.
If C1,C2 are two concentric circles with centre at z=a and radii r1 and r2 (r1<r2) and if f(z) is analytic inside and on the circles and within the annulus between C1 andC2, then for any z in the annulus, we have ,where and where C is any circle lying
between C1 and C2 with centre at z=a for all n and the integration being taken in positive direction.
12. / Obtain the Laurent expansion of the function in the neighborhood of its singular point. Hence find the residue at that point.
z=0 is a pole of order 2becomes
Residue of f(z) = coefficient of =1
13. / Expand f(z)=in the region Using this result, expand and
tan-1z in powers of z.
By using binomial series expansions,
=…….(1)
----- (2)
If f(z)=tan-1z, then f’(z)=
Hence By integrating (2) w.r.to z tan-1z=
14. / State Cauchy’s integral formula for nth derivative of an analytic function.
If f(z) is analytic inside and on a simple closed curve C and z=a is any interior point or the region R enclosed by C, then
and in general ,the integration
around C being taken in anti-clockwise direction.
15. / Find the region of convergence to expand cos z in Taylor’s series.
Let f(z) = cos z
The region of convergence is
16. / State Cauchy’s residue theorem.
If f(z) is analytric inside a closed curve C except at a finite number of isolated singular points a1,a2,…an inside C, then =(sum of the residues of f(z) at these singular points).
17. / Find the poles and residues of .
Poles are 1,2
[Res of f(z)]z=1=
[Res of f(z)]z=2=
18. / Find the residue of the function at a simple pole.
Given . Here z = 1 is a simple pole
[Resf(z)]z=1 = 1
19. / Define essential singularity with an example
If the principal part contains an infinite number of non zero terms, then z=z0 is known as a essential singularity.
has z=0 as an essential singularity.
Since f(z) is an infinite series of negative powers of z.
20. / Define removable singularity with an example.
If the principal part of f(z) contains no terms,That is bn=0 for all n, then the singularity z=z0 is known as the removable singularity of f(z).
Example, =1-
There is no negative power of z.
Z=0 is a removable singularity.
PART B
1. / Using Cauchy’s integral formula, find , where Cis.
2. / Using Cauchy’s integral formula, evaluate, where Cis
3. / Expand f(z) = in a Laurent’s series expansion for and .
4 / Obtain the Laurent’s series expansion for the function f(z) = in
and .
5. / Find the residues of f(z)= at its isolated singularities using Laurent’s
series expansion.
6 / Using Cauchy’s residue theorem evaluate, where Cis
7. / Evaluate, where Cisusing Cauchy’s residue theorem
8 / Evaluate , using contour integration.
9. / Evaluate , using contour integration.
10. / Show that, using contour integration.
UNIT- V LAPLACE TRANFORM
1. / State the conditions under which Laplace transform of f(t) exists.
The Laplace transform of f(t) exists if
(i) f(t) is piecewise continuous in the closed interval(a,b) where a >0 .
(ii) f(t) is of exponential order.
2. / Find the Laplace transform of .
= = =
3. / If L[f(t)] = F(s), prove that L{f(t/5)}= 5 F(5s).
=
put 5du = dt
= = = 5 F(5s).
4 / Find the Laplace transform of unit step function.
The unit step function is defined as
and L[ua(t)] =
5. / Find the Laplace transform of f(t) = cos2 3t.
L[cos2 3t] = = = .
6. / Does exist?
does not exist.
7. / Obtain the Laplace transform of sin2t – 2tcos2t in the simplified form.
=
=
=
=
8. / Find .
=
=
=e-t
= e-t (cos t + sin t)
9. / What is the Laplace transform of f(t) = f(t +10), 0< t < 10?
Given thatf(t) is a periodic function with period 10
L{f(t)} =
Put p=10, L{f(t)} =
10. / If L{f(t)} = , findthe value of .
= = = =
11. / Find
=
= - 2
= – =
12. / Find the Laplace transform sin32t
L[sin32t] = ¼ L[3sin2t – sin6t] = ¾ L[sin2t] – ¼ L[sin6t]
= - = - .
13. / Find
Let F(s) =
F(s) = =
;
14. / Solve using Laplace transform given that y(0)=0.
Taking L.T. on both sides, we get L[y] + L[y] = L[e-t]
(s+1) L[y] =
L[y] =; y = = t e-t.
15. / Give an example for a function that do not have Laplace transform.
Consider f(t)=, since , hence is not of exponential order function. Hence f(t)=, does not have Laplace transform.
16. / Can F(s)= be the transform of some f(t)?
0
Hence F(s) cannot be Laplace transform of f(t).
17. / Evaluate .
Let
= L[sint] L[cost] (by convolution theorem)
=
=
18. / Give an example for a function having Laplace transform but not satisfying the continuity condition.
f(t) = has Laplace transform even though it does not satisfy the continuity condition. i.e. It is not piecewise continuous in (0,) as
19. / Define a periodic function and give examples
A function f(t) is said to be periodic function if f(t+p) =f(t) for all t. The least value of p> 0 is called the period of p.For example, sint and cost are periodic functions with period 2
20. / State the convolution theorem
The convolution of f(t) and g(t) is defined as and it is denoted by f(t) * g(t).
PART B
1. / Find the Laplace transform of f(t) = and f(t+2a) = f(t) for all t.
2. / Find the Laplace transform of f(t) = and f(t+2a) = f(t) for all t.
3. / Find the inverse Laplace transform of using convolution theorem.
4. / Find the Laplace transform of
5. / Verify initial and final value theorems for the function f(t) = 1 + e-t(sin t + cos t).
6. / Find the inverse Laplace transform of
7. / Using Laplace transform solve the differential equation y-3y-4y=2e-t with y(0) = y(0) = 1.
8. / Solve the equation y+ 9y=cos2t with y(0) =1 y () = –1.
9. / Evaluate using Laplace transform.
10 / Find the Laplace transform of (i) t2 e–tcost (ii) coshat cosat
1