Understanding Engineering Mathematics

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.

4. 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