whereλ>0 is some experimentally determined constant of proportionality, andT 0 is
the initial temperature. Solve this to give the temperature att>0.
3.In Applications, Chapter 2, we saw how the differential equation
dx
dt
=k(x−a)(x−b)(x−c)
typical of models of chemical reactions, describes the reaction between three gases
A,B,Cin a vapour deposition process. Find the general solution of the equation for
a number of examples of values ofk,a,b,c, starting with a simple case such as
k=a=b=1,c=0 and building up to more complicated examples. Also consider
various initial conditions. Study the nature of the solutions for the various values of
the parametersk,a,b,c.
- Theclassic application of differential equations is, of course, in Newtonian mechanics.
The differential equations arise from either thekinematicrelation dx/dt=v,the
velocity, or thedynamicalrelation (Newton’s second law)mdv/dt=md^2 x/dt^2 =
mass×acceleration=force. In general we have
d^2 x
dt^2
=f(x,y,
dy
dt
)
This reduces to a first order equation in a number of cases. For example, iff is
independent ofxwe have
dv
dt
=f(t,v)
or if it is independent oftwe can write
dv
dt
=
dx
dt
dv
dx
=v
dv
dx
=f(x,v)
With sufficiently complex force laws, we can obtain a variety of differential equations
sufficient to keep the most ardent mathematician satisfied – indeed, much of the modern
theory of nonlinear differential equationisdynamics, and you will find books on
‘dynamical systems’ shoulder to shoulder with textbooks on differential equations on
the library bookshelves.
A particle falling vertically under gravity, subject to a resistance proportional to its
velocity,v, satisfies the equation of motion
m
dv
dt
=mg−kv
wheremis its mass,gthe acceleration due to gravity andka positive constant. Solve
this equation and interpret the motion. Assume the initial velocity isv=v 0 att=0.
5.Many complicated physical, biological, and commercial situations can be modelled
by a system of interconnected units or ‘compartments’ between which some quantity
flows or is communicated. Examples include the distribution of drugs in various parts