Understanding Engineering Mathematics

is described by thedifferential equation

dx

dt

=k(x−a)(x−b)(x−c)

wherexis the number ofABCmolecules anda,b,care the initial concentrations of

theA,B,Cgases respectively.kis a constant depending on the rate of the chemical

reaction. Such equations are typical of a chemical reaction. In such a tri-molecular first

order reaction thex−afactor, for example, models the fact that the rate of production

of the new molecule is proportional (14

➤

) to the current amount of each component

substance available. Asxapproaches the smallest of thea,b,cvalues the rate of

the reaction will slow down, until it stops completely when there is no more of that

constituent left. The differential equation will be solved in Chapter 15 (➤477) and a

crucial step in the solution is the splitting of

1

(x−a)(x−b)(x−c)

into partial fractions. Do this for the case ofa=1,b=2,c=3.

Obtain an expression for the partial fractions in the case whena,b,chave general

values.

6.In the toss of a fair coin the probability of a head is^12 and the same for a tail. Suppose
we toss such a coin three times. We can get 3H, 2H1T, 1H2T, and 3T. Write down the
probability of each possibility. Show that the sum of these probabilities can be written
( 1
2 +

1

2

) 3

i.e. as a binomial expansion. In general, if we havenindependent ‘trials’, each resulting

in a ‘success’ with probabilitypand a ‘failure’ with probabilityq= 1 −pthen the

probability ofs‘successes’ is given by the binomial term

p(s)=nCsps( 1 −p)n−s

Use the binomial theorem to show that the sum of all these probabilities, for all possi-

bilities, is 1.p(s)defines thebinomial distribution, one of the most important in

statistics.

Answers to reinforcement exercises

2.3.1 Multiplication of linear expressions

A.(i), (ii), (iv), (v), (viii), (ix), (xii), (xiv) are algebraic expressions, the rest are either
not algebraic, such as sinx−3cosx, or are equations such as 2t+ 1 =0 rather than
expressions.

B. (iii), (vii), (xi), (xv) are algebraic equations.

C. (i) a+b(c+d) (ii) (a+b)c+d (iii) (a+b)(c+d)

(iv) a−b(c+d) (v) (a−b)c−d (vi) (a−b)c+d

(vii) x^2 − 3 (x+ 4 ) (viii) a+b(c+bd) (ix) (a+bc+b)d

(x) (a−b)(c−d) (xi) a−b(c−d) (xii) (x^2 − 3 )x+ 4