university, bhopal assignment question paper - Madhya Pradesh ...

M P BHOJ (OPEN) UNIVERSITY, BHOPAL
ASSIGNMENT QUESTION PAPER
2009-10
CLASS : M.Sc. Final SUBJECT: Mathematics
Paper - II - Partial differential Equations & Mechanics
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Q1. Find the solution of ∂ 2 u/∂x 2 + 3∂ 2 u/∂x∂y + 2∂ 2 u/∂y 2 = 0.
First Assignment Max Marks - 30
Q2. Solve y′′′ - 3y′ + 3y′ - y = t 2 e t , where y(0) = 1 , y′(o) = 0 & y′′′(0) = -2, by using Laplace
transform, where y′ = ∂y/∂x.
Q3. A particle of mass m moves along the x – axis & is attracted towards origin O with a force
numerically equal to kx, k > 0. A damping force given by β dx/ dt , β > 0 also acts. Discus the
motion treating all cases, assuming that X(0) = X0 , Y′ (0) = V0.
Q4. A tightly streached flexible string has its end fixed at x = 0 & x = l. At time t = 0 the string has
given a shape defined by F(x) = µx ( l – x), where µ is a constant and then released. Find the
displacement of any point x of the string at any time t > 0.
Q5. Find the solution of the wave equation ∂ 2 y/∂t 2 = c 2 ∂ 2 y/∂x 2 Such that y = p0 cos pt (p0 is a
constant) when x = l and y = 0 when x = 0.
Second Assignment Max Marks - 30
Q1. Solve ∂u/∂t = 2 ∂ 2 u/∂x 2 , u(0, t) = 0 , u(5 ,t) = 6, u(x, 0) = 10sin4πx.
Q2. Solve ∂ 2 z/∂x 2 + ∂ 2 z/∂y 2 = cosmx cosny
Q3 A string is stretched between the fixed points (0, 0) & (1, 0) & released from rest from the
position y(x,0) = Asin2πx.. Find the displacement y(x, t).
Q4. Find the temperature u(x, t) in a bar of length l, which is perfectly insulated whose ends are
already kept at temp. zero & initial temp. is
x , 0 < x < l/2
F(x) = l-x, l/2 < x