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Dialogue: 0,0:00:00.07,0:00:03.17,Default,,0000,0000,0000,,So here is the connection with differential equations:
Dialogue: 0,0:00:03.17,0:00:09.33,Default,,0000,0000,0000,,The exponential function is the unique solution to the following differential equation:
Dialogue: 0,0:00:09.33,0:00:19.47,Default,,0000,0000,0000,,The derivative of Y of X, with respect to X, equals Y of X, for all X,
Dialogue: 0,0:00:19.47,0:00:28.47,Default,,0000,0000,0000,,and the initial value, Y of zero, equals one. This is about as simple as it can get with differential equations.
Dialogue: 0,0:00:28.47,0:00:35.40,Default,,0000,0000,0000,,We're looking for a function, Y, that is its own derivative, and starts with a value of one.
Dialogue: 0,0:00:35.40,0:00:40.30,Default,,0000,0000,0000,,It's clear now that the exponential function is a solution to that differential equation,
Dialogue: 0,0:00:40.30,0:00:46.40,Default,,0000,0000,0000,,but as most other differential equations that are of interest, also this differential equation is so well-behaved
Dialogue: 0,0:00:46.40,0:00:52.43,Default,,0000,0000,0000,,that the exponential function is the unique solution of this differential equation.