DIFFERENTIAL EQUATIONS 
339 
In a similar manner we can iind a second solution of (1). Sub 
tract e™ 2 * from e m ' x ; tlie result is 
2 V— i e~ ax sin yx. 
It is true that this result is imaginary, but only through the 
presence of an imaginary constant factor, 2 V — 1. Suppress this 
factor and consider the function 
(7) 
y = e~ ax sin yx. 
On substituting this function into the given differential equation, 
as was done with the function (6), we find that (7) also satisfies that 
equation, and thus we have the best of all proofs that (7) is a solu 
tion — that of direct substitution. • For, a function that satisfies a 
differential equation is a solution, no matter how obscure its origin • 
and one that does not satisfy it is not a solution, no matter how 
illustrious its pedigree may seem to have been. 
We have introduced this bit of eighteenth century mathematics 
partly to give a motif for the two solutions (6) and (7) ; partly to 
show how mathematicians obtained true results from working with 
V— 1, long before they knew how to define that number. They 
divined its importance, but they did not yet have the vision to give 
it existence through definition, as is seen from a remark of Leib 
niz in the year 1702*: “Die imaginären Zahlen sind eine feine und 
wunderbare Zuflucht des göttlichen Geistes, beinahe ein Ampliibium 
zwischen Sein und Nichtsein.” 
Equations of the n-th Order. The method can be extended at once 
to the equation 
(») 
dx n dx n ~ l 
+ «»y = o, 
where the a’s are constants. On substituting y = e mx we find that 
this function is a solution provided m is a root of the algebraic 
equation 
(9) m" + cqm" -1 + ••• + a n = 0. 
If this equation has n real and distinct roots, m 1? •••, m B , the 
general solution of (8) will be 
(10) y=c ie"i*+ ••• + c n e m “ x . 
If one of the roots of (9) is imaginary, 
m l =p + q V— 1, 
* Klein, Elementarmathematik vom höheren Standpunkte aus, 3d ed., vol. I, 
p. 61.