Math Answers

Use the power series method to obtain one solution of the following ODE:
3 0
2
x y ′′ + y′ − xy

A mass of 2 kg is fixed to an end of a spring with spring constant k = 128 Nm−1
and the
system is placed inside a fluid. It is set into vibration from its equilibrium position with
an initial speed of 0.6 ms−1
. If the damping force due to the fluid is 40v(t) N where v is
the instantaneous speed of the mass, determine the position of the mass as a function of
time t.

A paratrooper weighing 80 kg jumps with zero velocity from an aeroplane at a height of
3000 m. The air resistance encountered by the paratrooper is 2 R t)( =15v t)( N where v(t)
is the velocity of the paratrooper at time t. Calculate the time required by the paratrooper
to land and the velocity at landing.

Solve the initial value problem:
2 2 ,0 )0( ,2 )0( 1
2
2
+ + y = y = y′ =
dx
dy
dx
d y

Solve the following ordinary differential equations:
i)

sin [ ] ( ) ln )(cos ) 0
1
ydx + x y + y dy =
x

Trace the curve y2(x2- 9) =x4 by stating all the properties used in tracing.

Find the radius and the center of the circular section of the sphere |r| = 4 cut off by the
plane r·(2i−j+ 4k) = 3

Which of the following are binary operations. Justify your answer.
i) The operation · defined on Q by a·b = a(b− 1).
ii) The operation · defined on [0,π] by x·y = cosxy.
Also, for those operations which are binary operations, check whether they are
associative and commutative.

Find the nature of the conic 2x^2 + xy− y^2− 6 = 0.