We now construct the exponential function as the solution to the differential equation or initial value problem
- for t>0
which we compute by time stepping (expressing ).
Note that the initial value problem for t>0 with , where C is any number, is solved by , as an expression of the linearity of the differential equation allowing multiplication by any number, because , or computationally if .
We can extend to by time stepping backwards in time.
The exponential function is the unique function which is equal to the derivative of itself as a form of complete self satisfaction.
- Play with the code changing e g into dx =alpha*x*dt with alpha a given number.
- Show that by construction by using the linearity of the differential equation (not the solution) discussed above. Hint: Show that satisfies for t > 0 with , and so is equal to the multiple of .