Mathematical Methods for Physics and Engineering : A Comprehensive Guide

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS

If the beam is only slightly bent, so that (dy/dx)^2 1, wherey=y(x)isthe

downward displacement of the beam atx, show that the beam profile satisfies

the approximate equation

d^2 y

dx^2

=−

Wx

2 K

.

By integrating this equation twice and using physically imposed conditions on
your solution atx=0andx=L/2, show that the downward displacement at
the centre of the beam isWL^3 /(48K).
15.4 Solve the differential equation

d^2 f

dt^2

+6

df

dt

+9f=e−t,

subject to the conditionsf=0anddf/dt=λatt=0.
Find the equation satisfied by the positions of the turning points off(t)and
hence, by drawing suitable sketch graphs, determine the number of turning points
the solution has in the ranget>0if(a)λ=1/4, and (b)λ=− 1 /4.
15.5 The functionf(t) satisfies the differential equation

d^2 f

dt^2

+8

df

dt

+12f=12e−^4 t.

For the following sets of boundary conditions determine whether it has solutions,

and, if so, find them:

(a) f(0) = 0,f′(0) = 0,f(ln

√

2) = 0;

(b)f(0) = 0,f′(0) =− 2 ,f(ln

√

2) = 0.

15.6 Determine the values ofαandβfor which the following four functions are
linearly dependent:

y 1 (x)=xcoshx+sinhx,

y 2 (x)=xsinhx+coshx,

y 3 (x)=(x+α)ex,

y 4 (x)=(x+β)e−x.

You will find it convenient to work with those linear combinations of theyi(x)
that can be written the most compactly.
15.7 A solution of the differential equation

d^2 y

dx^2

+2

dy

dx

+y=4e−x

takes the value 1 whenx= 0 and the valuee−^1 whenx= 1. What is its value
whenx=2?
15.8 The two functionsx(t)andy(t) satisfy the simultaneous equations

dx

dt

− 2 y=−sint,

dy

dt

+2x=5cost.

Find explicit expressions forx(t)andy(t), given thatx(0) = 3 andy(0) = 2.

Sketch the solution trajectory in thexy-plane for 0≤t< 2 π, showing that

the trajectory crosses itself at (0, 1 /2) and passes through the points (0,−3) and

(0,−1) in the negativex-direction.