Gauhati University 2007 Previous : thetics--V - Question Paper

2007
MATHEMATICS
FIFTH PAPER
(Differential Equations, Numerical Analysis and Computer Programming)
Full Marks: 80
Time: three hours
The figures in the margin indicate full marks for the ques.
PART-A
(Objective-type Questions)
(Marks: 32)
1. (a) Justify that y =xm is a particular integral of
If m(m-1)+Pmx+Qx2 = 0
(b) provide the geometrical interpretation of
(c) discuss the role of indicial formula in finding the power series solution of a
second-order ordinary differential formula.
(d) When x = x0 is called an ordinary point, singular point, regular singular point
and irregular singular point of the differential formula
(e) Expand the Wronskian W [y1, y2 and y3] (x) if y1, y1, and y3 are 3 linearly
independent solutions.
(f) State how the linear dependency or independency of solutions for a differential
formula depends on the Wronskian.
(g) State when the differential formula
has unique solution.
0 2
2
+ +Qy =
dx
dy
P
dx
d y
R
dz
Q
dy
P
dx = =
0? 2
2
+ +Qy =
dx
dy
P
dx
d y
0 0 f (x, y), y(x ) y
dx
dy = =
(h) If y1 is a particular solution of the Riccati equation, then mention the way of
finding the general solution. 2×8=16
2. (a) Under what situation Newton’s divided difference formula is adopted?
(b) discuss the difference ranging from the Newton’s interpolation and Lagrange’s
interpolation formula.
(c) Write Simpson’s
rd rule and trapezoidal rule. discuss the efficiency of one
over the other.
(d) discuss the role of the 2 ‘*’ in learn (*,*) statement in FORTRAN
language.
(e) accurate the program segment:
DO 10 I = 1, 10
I = five + I
10 continue on
(f) Write log10(x) and loge(x) in FORTRAN.
(g) Write FORTRAN statement and logical if statement in FORTRAN.2×8=16
(h) Write arithmetic if statement and logical if statement in FORTRAN. 2×8=16
PART-B (Subjective-type Questions)
Section-I
(Differential Equations)
(Marks: 24)
ans any 2 ques.
3. ans any three: 4×3=12
(a) Show that y=x is a part of the CF of
and hence solve it.
(b) Solve
(c) Solve
by removing the 1st derivative.
xy x
dx
dy
x
dx
d y - two + =
2
2
( two )
2
1
2
2
2 2
2 ( 2)
x x
x y e
dx
dy
x
dx
d y + - + + =
x y x x
dx
dy
x
dx
d y two 3
2
2
- cot - sin . = cos - cos
(d) Apply the method of variation of parameters to solve
4. ans part (a) and any 1 of (b), (c):
(a) Show that x=0 is an ordinary point for the differential formula
And hence solve it in powers of x. 7
(b) Solve the simultaneous equations 5
(c) Show that the condition of integrability is satisfied by the formula
Z(z-y)dx + (z+x)zdy +x(x+y)dz = 0 and solve it. 5
5. (a) Establish the subsequent outcome : 7
The general solution of the linear partial differential formula Pp + Qq = R is
f(u,v)=0 where f is an arbitrary function and u(x, y, z) = c1 and v(x, y, z) = c2 form
a solution of the equations
(b) Use Charpit’s method to obtain a complete integral of the formula
2zx-px2-2qxy+pq = 0 5
Section-II
(Numerical Analysis and Computer Programming)
(Marks: 24)
ans any 2 ques.
6. (a) The population of a country in census are as under. Estimate the population for the
year 1925: 6
Year (x) : 1891 1901 1911 1921 1931
Population (y) : 46 66 81 93 101
(in thousand)
ex
y
dx
d y
+
- =
1
2
2
2
2 0
2
2
- - y =
dx
dy
x
dx
d y
- seven + = 0; - 2x - 5y = 0
dt
dy
x y
dt
dx
R
dz
Q
dy
P
dx = =
(b) Derive the Newton’s divided difference interpolation formula. Mention when this
formula is used. 5+1=6
7. (a) Show that the rate of convergence of the Newton-Raphson method is quadratic.6
(b) Use the subsequent data to obtain f ' (5): 6
x : 0 two three four seven 9
f(x) : four 26 58 112 466 922
8. (a) Write a program in FORTRAN to obtain the largest and the smallest of 50 provide
numbers.
6
(b) Write a program in FORTRAN to solve the formula
x5 + 3cos(x)-2 tan(x) = 0
Using the algorithm of Newton-Raphson method. 6

Earning:   Approval pending.