Firk F.W.K. Essential Physics. Part 1. Relativity, Particle Dynamics, Gravitation, and Wave Motion
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O R D I N A R Y D I F F E R E N T I A L E Q U A T I O N S 193
v + x(dv/dx) = (1 + v
2
)/2
therefore
2xdv = (1 + v
2
– 2v)dx.
Separating the variables
2dv/(v – 1)
2
= dx/x., and this can be integrated.
Linear Equations
The equation
dy/dx + M(x)y = N(x)
is said to be linear and of the 1st order. An example of such an equation is
dy/dx + (1/x)y = x
2
.
This equation can be solved by introducing the integrating factor, x, so that
x(dy/dx) + y = x
3
,
therefore
(d/dx)(xy) = x
3
,
giving
xy = x
4
/4 + constant.
In general, let R be an integrating factor, then
R(dy/dx) + RMy = RN,
in which case, the left-hand side is the differential coefficient of some product with a first
term R(dy/dx). The product must be Ry! Put, therefore
R(dy/dx) + RMy = (d/dx)(Ry) = R(dy/dx) + y(dR/dx).
O R D I N A R Y D I F F E R E N T I A L E Q U A T I O N S 194
Now,
RMy = y(dR/dx),
which leads to
∫M(x)dx = ∫dR/R = lnR,
or
R = exp{∫M(x)dx}.
We therefore have the following procedure: to solve the differential equation
(dy/dx) + M(x)y = N(x),
multiply each side by the integrating factor exp{∫M(x)dx}, and integrate. For example, let
(dy/dx) + (1/x)y = x
2
,
so that ∫M(x)dx = ∫(1/x)dx = lnx and the integrating factor is exp{lnx} = x:. We
therefore obtain the equation
x(dy/dx) + (1/x)y = x
3
,
deduced previously on intuitive grounds.
Linear Equations with Constant Coefficients.
Consider the 1st order linear differential equation
p
0
(dy/dx) + p
1
y = 0, where p
0
, p
1
are constants.
Writing this as
p
0
(dy/y) + p
1
x = 0,
we can integrate term-by-term, so that
p
0
lny + p
1
x = constant,
O R D I N A R Y D I F F E R E N T I A L E Q U A T I O N S 195
therefore
lny = (–p
1
/p
0
)x + constant
= (–p
1
/p
0
)x + lnA, say
therefore
y = Aexp{(–p
1
/p
0
)x}.
Linear differential equations with constant coefficients of the 2nd order occur often in
Physics. They are typified by the forms
p
0
(d
2
y/dx
2
) + p
1
(dy/dx) + p
2
y = 0.
The solution of an equation of this form is obtained by following the insight gained in
solving the 1st order equation!. We try a solution of the type
y = Aexp{mx},
so that the equation is
Aexp{mx}(p
0
m
2
+ p
1
m + p
2
) = 0.
If m is a root of
p
0
m
2
+ p
1
m + p
2
= 0
then y = Aexp{mx} is a solution of the original equation for all values of A.
Let the roots be α and β. If α ≠ β there are two solutions
y = Aexp{αx }and y = Bexp{βx.}.
If we put
y = Aexp{αx} + Bexp{βx}
in the original equation then
O R D I N A R Y D I F F E R E N T I A L E Q U A T I O N S 196
Aexp{αx}(p
0
α
2
+ p
1
α + p
2
) + Bexp{βx}(p
0
β
2
+ p
1
β + p
2
) = 0,
which is true as α and β are the roots of
p
0
m
2
+ p
1
m + p
2
= 0, (called the auxilliary equation)
The original equation is linear, therefore the sum of the two solutions is, itself, a (third)
solution. The third solution contains two arbitrary constants (the order of the equation),
and it is therefore the general solution.
As an example of the method, consider solving the equation
2(d
2
y/dx
2
) + 5(dy/dx) + 2y = 0.
Put y = Aexp{mx }as a trial solution, then
Aexp{mx}(2m
2
+ 5m + 2) = 0, so that
m = –2 or –1/2, therefore the general solution is
y = Aexp{–2x} + Bexp{(–1/2)x}.
If the roots of the auxilliary equation are complex, then
y = Aexp{p + iq} + Bexp{p – iq},
where the roots are p ± iq ( p, q ∈ R).
In practice, we write
y = exp{px}[Ecosqx + Fsinqx]
where E and F are arbitrary constants.
For example, consider the solution of the equation
d
2
y/dx
2
– 6(dy/dx) + 13y = 0,
therefore
O R D I N A R Y D I F F E R E N T I A L E Q U A T I O N S 197
m
2
– 6m + 13 = 0,
so that
m = 3 ± i2.
We therefore have
y = Aexp{(3 + i2)x} + Bexp{3 – i2)x}
= exp{3x}(Ecos2x + Fsin2x).
The general solution of a linear differential equation with constant coefficients is the sum
of a particular integral and the complementary function (obtained by putting zero for
the function of x that appears in the original equation).
BIBLIOGRAPHY
Those books that have had an important influence on the subject matter and the
style of this book are recognized with the symbol *. I am indebted to the many authors
for providing a source of fundamental knowledge that I have attempted to absorb in a
process of continuing education over a period of fifty years.
General Physics
*Feynman, R. P., Leighton, R. B., and Sands, M., The Feynman Lectures on Physics,
3 vols., Addison-Wesley Publishing Company, Reading, MA (1964).
*Joos, G., Theoretical Physics, Dover Publications, Inc., New York, 3rd edn (1986).
Lindsay, R. B., Concepts and Methods of Theoretical Physics, Van Nostrand Company,
Inc., New York (1952).
Mathematics
Armstrong, M. A., Groups and Symmetry, Springer-Verlag, New York (1988).
*Caunt, G. W., An Introduction to Infinitesimal Calculus, The Clarendon Press, Oxford
(1949).
*Courant R., and John F., Introduction to Calculus and Analysis, 2 vols., John Wiley &
Sons, New York (1974).
Kline, M., Mathematical Thought from Ancient to Modern Times, Oxford University
Press, Oxford (1972).
B I B L I O G R A P H Y 199
*Margenau, H., and Murphy, G. M., The Mathematics of Physics and Chemistry, Van
Nostrand Company, Inc., New York, 2nd edn (1956).
Mirsky, L., An Introduction to Linear Algebra, Dover Publications, Inc., New York (1982).
*Piaggio, H. T. H., An Elementary Treatise on Differential Equations, G. Bell & Sons,
Ltd., London (1952).
Samelson, H., An Introduction to Linear Algebra, John Wiley & Sons, New York (1974).
Stephenson, G., An Introduction to Matrices, Sets and Groups for Science Students,
Dover Publications, Inc., New york (1986).
Yourgrau, W., and Mandelstam, S., Variational Principles in Dynamics and Quantum
Theory, Dover Publications, Inc., New York 1979).
Dynamics
Becker, R. A., Introduction to Theoretical Mechanics, McGraw-Hill Book Company, Inc.,
New York (1954).
Byerly, W. E., An Introduction to the Use of Generalized Coordinates in Mechanics and
Physics, Dover Publications, Inc., New York (1965).
Kilmister, C. W., Lagrangian Dynamics: an Introduction for Students, Plenum Press, New
York (1967).
*Ramsey, A. S., Dynamics Part I, Cambridge University Press, Cambridge (1951).
*Routh, E. J., Dynamics of a System of Rigid Bodies, Dover Publications, Inc., New York
(1960).
B I B L I O G R A P H Y 200
Whittaker, E. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies,
Cambridge University Press, Cambridge (1961). This is a classic work that goes
well beyond the level of the present book. It is, nonetheless, well worth
consulting to see what lies ahead!
Relativity and Gravitation
*Einstein, A.., The Principle of Relativity, Dover Publications, Inc., New York (1952). A
collection of the original papers on the Special and General Theories of Relativity.
Dixon, W. G., Special Relativity, Cambridge University Press, Cambridge (1978).
French, A. P., Special Relativity, W. W. Norton & Company, Inc., New York (1968).
Kenyon, I. R., General Relativity, Oxford University Press, Oxford (1990).
Lucas, J. R., and Hodgson, P. E., Spacetime and Electromagnetism, Oxford University Press,
Oxford (1990).
*Ohanian, H. C., Gravitation and Spacetime, W. W. Norton & Company, Inc., New York
(1976).
*Rindler, W., Introduction to Special Relativity, Oxford University Press, Oxford, 2nd edn
(1991).
Rosser, W. G. V., Introductory Relativity, Butterworth & Co. Ltd., London (1967).
Non-Linear Dynamics
*Baker, G. L., and Gollub, J. P., Chaotic Dynamics, Cambridge University Press,
Cambridge (1991).
Press, W. H., Teukolsky, S. A., Vetterling W. T., and Flannery, B. P., Numerical Recipes in
B I B L I O G R A P H Y 201
C, Cambridge University Press, Cambridge 2nd edn (1992).
Waves
Crawford, F. S., Waves, (Berkeley Physics Series, vol 3), McGraw-Hill Book Company, Inc.,
New York (1968).
French, A. P., Vibrations and Waves, W. W. Norton & Company, Inc., New York (1971).
General reading
Bronowski, J., The Ascent of Man, Little, Brown and Company, Boston (1973).
Calder, N., Einstein’s Universe, The Viking Press, New York (1979).
Davies, P. C. W., Space and Time in the Modern Universe, Cambridge University Press,
Cambridge (1977).
Schrier, E. W., and Allman, W. F., eds., Newton at the Bat, Charles Scribner’s Sons, New
York (1984).