1
00:00:00,067 --> 00:00:03,166
So here is the connection with differential equations:
2
00:00:03,167 --> 00:00:09,332
The exponential function is the unique solution to the following differential equation:
3
00:00:09,333 --> 00:00:19,466
The derivative of Y of X, with respect to X, equals Y of X, for all X,
4
00:00:19,467 --> 00:00:28,466
and the initial value, Y of zero, equals one. This is about as simple as it can get with differential equations.
5
00:00:28,467 --> 00:00:35,399
We're looking for a function, Y, that is its own derivative, and starts with a value of one.
6
00:00:35,400 --> 00:00:40,299
It's clear now that the exponential function is a solution to that differential equation,
7
00:00:40,300 --> 00:00:46,399
but as most other differential equations that are of interest, also this differential equation is so well-behaved
8
00:00:46,400 --> 00:00:52,433
that the exponential function is the unique solution of this differential equation.