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Here we are trying to obtain the value of y(2) given y(0) = 1. With the Soln:
Final
field highlighted, press @SOLVE. You can check that a solution takes about
6 seconds, while in the previous first-order example the solution was almost
instantaneous. Press $ to cancel the calculation.
This is an example of a stiff ordinary differential equation
. A stiff ODE is one
whose general solution contains components that vary at widely different rates
under the same increment in the independent variable. In this particular case,
the general solution, y(t) = 1+ t +C⋅e
100t
, contains the components ‘t’ and
‘C⋅e
100t
’, which vary at very different rates, except for the cases C=0 or C≈0
(e.g., for C = 1, t =0.1, C⋅e
100t
=22026).
The calculator’s ODE numerical solver allows for the solution of stiff ODEs by
selecting the option
_Stiff in the SOLVE Y’(T) = F(T,Y) screen. With this
option selected you need to provide the values of ∂f/∂y and ∂f/∂t. For the case
under consideration ∂f/∂y = -100 and ∂f/∂t = 100.
Enter those values in the corresponding fields of the
SOLVE Y’(T) = F(T,Y)
screen:
When done, move the cursor to the
Soln:Final field and press @SOLVE. This
time, the solution in produced in about 1 second. Press @EDIT to see the
solution: 2.9999999999, i.e., 3.0.
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