HP 48gII Graphing Calculator Manual do Utilizador descarregar pdf (Página 546)

100

101

Page 16-68

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

, 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

(‚ Ï˜ @@@OK@@@) to:

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