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Numerical solution for stiff first-order ODE
Consider the ODE: dy/dt = -100y+100t+101, subject to the initial condition
y(0) = 1.
Exact solution
This equation can be written as dy/dt + 100 y = 100 t + 101, and solved
using an integrating factor, IF(t) = exp(100t), as follows (RPN mode, with CAS
set to Exact mode):
‘(100*t+101)*EXP(100*t)’
`
‘t’
`
RISCH
The result is ‘(t+1)*EXP(100*t)’.
Next, we add an integration constant, by using: ‘C’
`+
Then, we divide by FI(x), by using: ‘EXP(100*t)’
`/
.
The result is: ‘
((t+1)*EXP(100*t)+C)/EXP(100*t) ’, i.e., y(t) = 1+ t +C
⋅
e
100t
. Use
of the initial condition y(0) = 1, results in 1 = 1 + 0 + C
⋅
e
0
, or C = 0, the
particular solution being y(t) = 1+t.
Numerical solution
If we attempt a direct numerical solution of the original equation dy/dt = -
100y+100t+101, using the calculator’s own numerical solver, we find that
the calculator takes longer to produce a solution that in the previous first-order
example. To check this out, set your differential equation numerical solver
(
‚ Ϙ @@@OK@@@
) to: