Anna University Chennai 2007 M.Sc Computer Science Differentialwith Applications - Question Paper
Differential Equestions with Applications
(b) Prove that :
4
[xpJ (x)] = xp J (x) dx p p-1
and [x p J (x)] = - x -p J (x). dx p p+1
13. (a) Show that the equation :
x - y = x
and x2p + q = xz
are compatible and find their solution.
(b) Using Jacobis method, find complete integral
2 2
of the equation p x + q y = z.
14. (a) Find the solution of the equation
d2z 82 z
a2 a2=x_y-oxz o yz
(b) Obtain the solution, valid when x, y > 0, xy > 1 of the equation
32 z 1
such that
Name of the Candidate :
7 6 9 1 M.Sc. DEGREE EXAMINATION, 2007
( MATHEMATICS )
( FIRST YEAR )
( PAPER - IV )
540. DIFFERENTIAL EQUATIONS WITH APPLICATIONS
( New Regulations )
May ] [ Time : 3 Hours
Maximum : 100 Marks
PART-A (8x5 = 40)
Answer any EIGHT questions.
All questions carry equal marks.
1. Show that y = Cje2* + C2xe2x is the general solution of y"- 4y' + 4y = 0 on any interval. Find also the particular solution for which y (0) = 2 and y'(0) = 2.
2. Find the general solution of y" + xy' + y = 0 in terms of power series in x.
3. Prove that:
(Jo(x)) = -J1(x)
4. Using Gamma function, prove that:
(2n + 1)! _
5. Find the general integrals of the partial differential equation Z (xp - yq) = y2 - x2.
6. Find the equation of the system of surfaces which cut orthogonally the cones of the system
x2 + y2 + z2 = cxy.
7. If u = f(x + iy) + g(x - iy) where / and g are arbitrary, show that
82 u . 52 u
"x 2 "r 2 = 0.
5 xz 5 yz
8. Find particular integral of (D2 - D')Z - 2y - x2.
9. A gas is contained in a regid sphere of radius a. Show that if C is the velocity of sound in the gas, the frequencies of purely radical oscillations are
-i when z1? z2.....
cl
are positive roots of the equation tan z = z.
10. Derive the D'Ahemberts solution of the one dimensional wave equation.
PART - B (3 x 20 = 60)
Answer any THREE questions.
Each question carries equal marks.
11. (a) Find the particular solution of y"+y = sec x.
(b) For the differential equation
2x2y" + x (2x+l)y/ - y = 0. locate and classify its singular points in x axis.
12. (a) Find the general solution of the differential
equation
x(l-x)y" + (y-2xj y'+2y = 0 near the singular point x = 0.
z = o, p =
x + y
on the hyperbola xy = 1.
15. If the string is released from rest in the position y = f(x), show that the total energy of the string is
S
where
1
ks = T ' f(x)sin v7~ /dx
I q I
The mid-point of the string is pulled aside through a small distance and then released. Show that in the subsequent motion the fundamental mode
contributes t of the total energy.
71
z = o, p =
x + y
on the hyperbola xy = 1.
15. If the string is released from rest in the position y = f(x), show that the total energy of the string is
S
where
1
2 f (snx\
ks = T J f(x) sin ( j dx i 0 i
The mid-point of the string is pulled aside through a small distance and then released. Show that in the subsequent motion the fundamental mode
contributes t f the total energy. n
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| Earning: Approval pending. |
