The time scale exponential is used to solve time scale differential equations. On continuous time, the solution to the first order differential equation
![First Order DE](../Images/CTDE.jpg)
is
.
More generally, the solution to
is
.
This can be generalized to time scale equations of the form
.
The solution is
![DE](../Images/CTDE6.jpg)
where the time scale exponential is
![exp](../Images/CTDE7.jpg)
and the cylinder transformation is
![DE](../Images/CTDE8.jpg)
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