The definition of e^{x} as the solution of the 'initial value problem'...

y^{'}= y\ ,\ y(0)=1 (1)

... is [in principle...] perfecly acceptable. The 'initial condition' guarantees that y(x) is analytic in x=0 , so that we can write...

y(x)= \sum_{n=0}^{\infty} \frac{y^{(n)} (0)}{n!}\ x^{n} (2)

From (1) we can easily derive that \forall n\ y^{(n)} (0)=1 , so that is...

y(x)= e^{x}= \sum_{n=0}^{\infty} \frac{x^{n}}{n!} (3)

The 'more traditional' alternative is to define...

e^{x}= \lim_{n \rightarrow \infty} (1+\frac{x}{n})^{n} (4)

... and from (4) is possible to derive all the properties of e^{x} , including the fundamental identity...

\lim_{n \rightarrow \infty} (1+\frac{x}{n})^{n} = \lim_{n \rightarrow \infty}\sum_{k=0}^{n} \frac{x^{k}}{k!} (5)

... so that defintions (3) and (4) are fully equivalent...

Kind regards

\chi \sigma