Appendix B
Linear second-order

differential equation

The paraxial ray equation (2.138, 2.162, 2.239) are examples of
a more general linear second-order differential equation. We seek
a solution for a function y(x), which satisfies an equation of the
general form
d
2 y
dy
P (x)
+ Q(x)
+ R(x) y = S(x),
(B.1)
dx 2
dx
where P, Q, R, and S are known functions of x. We shall see in the
following that this applies directly to the problem of the chromatic
aberration.
In the case S = 0 the equation is designated homogeneous, and in
the case S � = 0 the equation is designated inhomogenous. The solution y h (x)of the homogenous equation can always be expressed as
a linear combination of two functions u 1 (x) and u 2 (x) as follows:
y h (x) = c 1 u 1 (x) + c 2 u 2 (x),
(B.2)
where c 1 and c 2 are constants, and where u 1 and u 2 are not constant multiples of one another. The paraxial ray equations (2.162,
2.239) for the transverse displacement v(z) = x(z) + i y(z) in the
rotated system is an example of just such a homogeneous equation.
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