1.  [50]

(a)  LEEDS1005 Find the solution of the following initial value problem

dx + x2 - 1 y = x - 1 ,    y = 1 at x = 0.

(b)  LEEDS1005 Let a > 0 be a constant. Consider the following differential equation

dy            y2

dx     ^a2 - x2

defined for x · (-a, a).  Identify the type of this differential equation and find its general solution.

Give the solution corresponding to the following initial value: y = -  at x =  .  Give an example of initial value for which the integration constant in the general solution is zero.

(c)  LEEDS1005 Find the solution of the following initial value problem

d2y         dy                                            dy

dx2           dx                                           dx

(d)  LEEDS1005 For the following homogeneous differential equation, given that y1 (x) = e北  is a solution, find the other independent solution y2 . Then, check explicitly that y1  and y2  are independent.

d2y                    dy

dx2                            dx

(e)  LEEDS1005 Consider the following homogeneous system of first order equations,

dx                      dy

dt                      dt

where a is a parameter.

i. Suppose a = -2. Give the general solution of this system in the form

x(t) = c1u(t) r1 + c2v(t) r2 ,

where u, v , r1 and r2 should be found and c1 , c2 are arbitrary constants. What is the nature of the critical point?

ii.  Find the range of values of a for which the critical point is a saddle.

2.  [30]

A particle of mass m is acted upon by gravity -mgk and linear resistance -mfvk where k is the unit vector directed upwards and f > 0 is a constant.  The particle is dropped with zero initial velocity from a height h above ground level which is chosen to coincide with z = 0.

(a)  LEEDS1005 What are the physical dimension of the coefficient f and its units?

Write down the equation of motion. By solving this equation with the given initial conditions, find the solution v(t) for the velocity as a function of time. Check that your answer has the correct physical dimension for a velocity.

(b)  LEEDS1005 Viewing the velocity as a function of z, solve the equation of motion

with the given initial conditions to find the relation between z and v . Deduce from this relation that the value v0  of the velocity when the particle hits the ground is the solution of the equation

v0 -  ln /1 + v0 、= h .                                     (1)

(c)  LEEDS1005  Under the assumption that f is very small, perform a Taylor series expansion to second order in (1) and obtain the value of v0  in this approximation in terms of g and h. Denote this velocity v1 .

Upon entering the ground with velocity v1 , the particle is subject to a constant friction force Fk where F > 0 is a constant. Calculate the distance d travelled by the particle into the ground in terms of m,g , h and F .

3.  [20]

A particle of mass m is acted on by an attractive central force f (r) = -  with respect to the origin O, where λ is a positive constant.  The particle is projected with initial speed v0  > 0 from a point at distance a > 0 from O and making an angle α =  with the radius vector.

(a)  LEEDS1005 Give the condition on v0  to ensure that the orbit of the particle is an

ellipse, in the form

v0  < vc

where the critical value vc  should be given in terms of λ and a.

(b)  LEEDS1005 Suppose that v0  = ′  .  Show that the orbit is a parabola (using

a simple criterion, you are not required to derive the equation for the trajectory). Find the distance of closest approach rmin  in terms of a.

(c)  LEEDS1005 Suppose now that v0  = vc .  Find the value of the total energy E0 and of the angular momentum L0 . Calculate the value of the eccentricity e of the orbit.

(d)  LEEDS1005 With the same initial conditions, find the new expression of the total energy E0  if the expression of the central force is modified to f (r) = -  .