Mathematical Methods for Physics and Engineering : A Comprehensive Guide

15.4 EXERCISES

15.3.6 Equations havingy=Aexas a solution

Finally, we note that if any general (linear or non-linear)nth-order ODE is

satisfied identically by assuming that

y=

dy

dx

=···=

dny

dxn

(15.88)

theny=Aexis a solution of that equation. This must be so becausey=Aexis

a non-zero function that satisfies (15.88).

Find a solution of

(x^2 +x)

dy

dx

d^2 y

dx^2

−x^2 y

dy

dx

−x

(

dy

dx

) 2

=0. (15.89)

Settingy=dy/dx=d^2 y/dx^2 in (15.89), we obtain

(x^2 +x)y^2 −x^2 y^2 −xy^2 =0,

which is satisfied identically. Thereforey=Aexis a solution of (15.89); this is easily
verified by directly substitutingy=Aexinto (15.89).

Solution method.If the equation is satisfied identically by making the substitutions

y=dy/dx=···=dny/dxntheny=Aexis a solution.

15.4 Exercises

15.1 A simple harmonic oscillator, of massmand natural frequencyω 0 , experiences
an oscillating driving forcef(t)=macosωt. Therefore, its equation of motion is
d^2 x
dt^2

+ω^20 x=acosωt,

wherexis its position. Given that att= 0 we havex=dx/dt= 0, find the
functionx(t). Describe the solution ifωis approximately, but not exactly, equal
toω 0.
15.2 Find the roots of the auxiliary equation for the following. Hence solve them for
the boundary conditions stated.

(a)

d^2 f

dt^2

+2

df

dt

+5f=0, withf(0) = 1,f′(0) = 0.

(b)

d^2 f

dt^2

+2

df

dt

+5f=e−tcos 3t, withf(0) = 0,f′(0) = 0.

15.3 The theory of bent beams shows that at any point in the beam the ‘bending
moment’ is given byK/ρ,whereKis a constant (that depends upon the beam
material and cross-sectional shape) andρis the radius of curvature at that point.
Consider a light beam of lengthLwhose ends,x=0andx=L, are supported
at the same vertical height and which has a weightWsuspended from its centre.
Verify that at any pointx(0≤x≤L/2 for definiteness) the net magnitude of
the bending moment (bending moment = force×perpendicular distance) due to
the weight and support reactions, evaluated on either side ofx,isWx/2.