Mathematical Methods for Physics and Engineering : A Comprehensive Guide

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS

verify that the golden mean is equal to the larger root of the recurrence relation’s
characteristic equation.
15.16 In a particular scheme for numerically modelling one-dimensional fluid flow, the
successive values,un, of the solution are connected forn≥1 by the difference
equation
c(un+1−un− 1 )=d(un+1− 2 un+un− 1 ),

wherecanddare positive constants. The boundary conditions areu 0 =0and
uM= 1. Find the solution to the equation, and show that successive values ofun
will have alternating signs ifc>d.
15.17 The first few terms of a seriesun, starting withu 0 ,are1, 2 , 2 , 1 , 6 ,−3. The series
is generated by a recurrence relation of the form

un=Pun− 2 +Qun− 4 ,

wherePandQare constants. Find an expression for the general term of the

series and show that, in fact, the series consists of two interleaved series given by

u 2 m=^23 +^134 m,

u 2 m+1=^73 −^134 m,

form=0, 1 , 2 ,....
15.18 Find an explicit expression for theunsatisfying

un+1+5un+6un− 1 =2n,

given thatu 0 =u 1 = 1. Deduce that 2n−26(−3)nis divisible by 5 for all
non-negative integersn.
15.19 Find the general expression for theunsatisfying

un+1=2un− 2 −un

withu 0 =u 1 =0andu 2 = 1, and show that they can be written in the form

un=

1

5

−

2 n/^2

√

5

cos

(

3 πn

4

−φ

)

,

where tanφ=2.
15.20 Consider the seventh-order recurrence relation

un+7−un+6−un+5+un+4−un+3+un+2+un+1−un=0.

Find the most general form of its solution, and show that:

(a) if only the four initial valuesu 0 =0,u 1 =2,u 2 =6andu 3 = 12, are specified,

then the relation has one solution that cycles repeatedly through this set of

four numbers;

(b) but if, in addition, it is required thatu 4 = 20,u 5 =30andu 6 = 42 then the

solution is unique, withun=n(n+1).

15.21 Find the general solution of

x^2

d^2 y

dx^2

−x

dy

dx

+y=x,

given thaty(1) = 1 andy(e)=2e.
15.22 Find the general solution of

(x+1)^2

d^2 y

dx^2

+3(x+1)

dy

dx

+y=x^2.