Number Theory: An Introduction to Mathematics

38 I The Expanding Universe of Numbers

Proposition 28 only guarantees the local existence of solutions, but this is in the
nature of things. For example, ifn=1, the unique solution of the differential equation

dx/dt=x^2

such thatx(t 0 )=ξ 0 >0isgivenby

x(t)={ 1 −(t−t 0 )ξ 0 }−^1 ξ 0.

Thus the solution is defined only fort<t 0 +ξ 0 −^1 , even though the differential equation
itself has exemplary behaviour everywhere.
To illustrate Proposition 28, taken=1andletE(t)be the solution of the (linear)
differential equation

dx/dt=x (3)

which satisfies the initial conditionE( 0 )=1. ThenE(t)is defined for|t| <R,
for someR>0. If|τ|<R/2andx 1 (t)=E(t+τ),thenx 1 (t)is the solution of
the differential equation (3) which satisfies the initial conditionx 1 ( 0 )=E(τ).But
x 2 (t)=E(τ)E(t)satisfies the same differential equation and the same initial condi-
tion. Hence we must havex 1 (t)=x 2 (t)for|t|<R/2, i.e.

E(t+τ)=E(t)E(τ). (4)

In particular,

E(t)E(−t)= 1 , E( 2 t)=E(t)^2.

The last relation may be used to extend the definition ofE(t), so that it is continuously
differentiable and a solution of (3) also for|t|< 2 R. It follows that the solutionE(t)
is defined for allt∈Rand satisfies theaddition theorem(4) for allt,τ∈R.
It is instructive to carry through the method of successive approximations explicitly
in this case. If we takex 0 (t)to be the constant 1, then

x 1 (t)= 1 +

∫t

0

x 0 (τ)dτ= 1 +t,

x 2 (t)= 1 +

∫t

0

x 1 (τ)dτ= 1 +t+t^2 / 2 ,

···.

By induction we obtain, for everyn≥1,

xn(t)= 1 +t+t^2 /2!+···+tn/n!.

Sincexn(t)→E(t)asn→∞, we obtain for the solutionE(t)the infinite series
representation

E(t)= 1 +t+t^2 /2!+t^3 /3!+···,

valid actually for everyt∈R. In particular,

e:=E( 1 )= 1 + 1 + 1 /2!+ 1 /3!+···= 2. 7182818 ....