Mathematical Methods for Physics and Engineering : A Comprehensive Guide

5.14 HINTS AND ANSWERS

5.33 If

I(α)=

∫ 1

0

xα− 1

lnx

dx, α >− 1 ,

what is the value ofI(0)? Show that

d

dα

xα=xαlnx,

and deduce that

d

dα

I(α)=

1

α+1

.

Hence prove thatI(α)=ln(1+α).
5.34 Find the derivative, with respect tox, of the integral

I(x)=

∫ 3 x

x

expxt dt.

5.35 The functionG(t, ξ) is defined for 0≤t≤πby

G(t, ξ)=

{

−costsinξ forξ≤t,

−sintcosξ forξ>t.

Show that the functionx(t) defined by

x(t)=

∫π

0

G(t, ξ)f(ξ)dξ

satisfies the equation

d^2 x

dt^2

+x=f(t),

wheref(t)canbeanyarbitrary (continuous) function. Show further thatx(0) =

[dx/dt]t=π= 0, again for anyf(t), but that thevalueofx(π) does depend upon

the form off(t).

[ The functionG(t, ξ) is an example of a Green’s function, an important

concept in the solution of differential equations and one studied extensively in

later chapters. ]

5.14 Hints and answers

5.1 (a) (i) 2xy , x^2 ; (ii) 2x, 2 y; (iii)y−^1 cos(x/y),(−x/y^2 )cos(x/y);
(iv)−y/(x^2 +y^2 ),x/(x^2 +y^2 ); (v)x/r, y/r, z/r.
(b) (i) 2y, 0 , 2 x; (ii) 2, 2 ,0; (v) (y^2 +z^2 )r−^3 ,(x^2 +z^2 )r−^3 ,−xy r−^3.
(c) Both second derivatives are equal to (y^2 −x^2 )(x^2 +y^2 )−^2.
5.3 2 x=− 2 y−x.Forg, both sides of equation (5.9) equaly−^2.
5.5 ∂^2 z/∂x^2 =2xz(z^2 +x)−^3 ,∂^2 z/∂x∂y=(z^2 −x)(z^2 +x)−^3 ,∂^2 z/∂y^2 =− 2 z(z^2 +x)−^3.
5.7 (0,0), (a/ 4 ,−a)and(16a,− 8 a). Only the saddle point at (0,0).
5.9 The transformed equation is 2(x^2 +y^2 )∂f/∂v= 0; hencefdoes not depend onv.
5.11 Maxima, equal to 1/8, at±(1/ 2 ,− 1 /2), minima, equal to− 1 /8, at±(1/ 2 , 1 /2),
saddle points, equalling 0, at (0,0), (0,±1), (± 1 ,0).
5.13 Maxima equal toa^2 e−^1 at (±a,0), minima equal to− 2 a^2 e−^1 at (0,±a), saddle
point equalling 0 at (0,0).
5.15 Minimum at (0,0); saddle points at (± 1 ,±1). To help with sketching the contours,
determine the behaviour ofg(x)=f(x, x).
5.17 The Lagrange multiplier method givesz=y=x/2, for a maximal area of 4.