The exponential function arises from solving fixed point problems in linear differential equations, specifically represented by the equation y' = y. This foundational concept leads to the general form c · e^{g(x)} as a modification of the exponential function. The defining property e^x · e^y = e^{x+y} illustrates the relationship between the multiplicative structure of curved phase spaces and the linearity of tangent spaces. Additionally, the connection between exponential functions and trigonometric functions is established through complex numbers, reinforcing the significance of the exponential function in differential equations.
PREREQUISITESMathematicians, students of differential equations, and anyone interested in the foundational concepts of exponential functions and their applications in solving differential equations.
That's a good question. I like to see it this way: The exponential function solves a fix point problem, namely ##y'=y##. A general differential equation is a kind of variation of it, ##y'=f(y)##, so it's plausible that a some sort of modification of the exponential function, ##c \cdot e^{g(x)}##, comes into play.TachyonLord said:For example, in linear differential equations, there might be these questions where we'd directly use e∫pdx as the integrating factor and then substitute it in this really cliche formula but I never really understood where it came from. Help ?
Thank you so much :)fresh_42 said:That's a good question. I like to see it this way: The exponential function solves a fix point problem, namely ##y'=y##. A general differential equation is a kind of variation of it, ##y'=f(y)##, so it's plausible that a some sort of modification of the exponential function, ##c \cdot e^{g(x)}##, comes into play.
Another point of view is the fundamental (and defining!) formula ##e^x\cdot e^y=e^{x+y}\,.## It translates the multiplicative structure of some curved phase space, which the solutions to a differential equation system are, into the linearity of its tangent spaces, or for short ##e^0=1## or ##e^{\operatorname{trace}}=\det##. In a differential equation system we have some statements about tangent spaces, i.e. something linear: ##x+y##, and we are searching for the curved solution to which these are the tangent spaces, i.e. ##x+y \mapsto f(x)\cdot f(y)\,.## And such a relation defines the exponential function. The next prominent functions are periodic ones, the trigonometric functions. However, a little detour to the complex numbers shows us, that they are, too, a combination of exponential functions.
So for me it is the property ##e^{\text{linear}} = \text{curved}\; , \;e^{\text{tangents}}=\text{flow}\,.##