Unformatted text preview: Continuous dynamic systems
Table 2.6 Maple input instructions for ﬁrstorder initial value problems
Problem Input instructions (i) dp
= p(a − bp), p(0) = p0
dt (ii) dn
= −λn, n(0) = n0
dt (iii) dy
= x2 − 2x + 1, y(0) = 1
dx dsolve({diff(p(t),t)=
p(t)*(ab*p(t)),p(0)=p0},p(t));
dsolve({diff(n(t),t)=
lambda*n(t),n(0)=n0},n(t));
dsolve({diff(y(x),x)=
x^22*x+1,y(0)=1},y(x)); are treated in a similar manner. If we have the initial value problem,
dy
= f ( y, t )
dt y(0) = y0 then the input instruction is
dsolve({diff(y(t),t)=f(y(t),t),y(0)=y0},y(t)); For example, look at table 2.6.
2.12.2 Secondorder equations Secondorder differential equations are treated in fundamentally the same way. If
we have the homogeneous secondorder differential equation
a dy
d2 y
+ b + cy = 0
2
dt
dt then the input instruction is
dsolve(a*diff(y(t),t$2)+b*diff(y(t),t)+c*y(t)=0,y(t)); If we have the nonhomogeneous secondorder differential equation
a d2 y
dy
+ b + cy = g(t)
2
dt
dt then the input instruction is
dsolve(a*diff(y(t),t$2)+b*diff(y(t),t)+c*y(t)
=g(t),y(t)); Of course, the solutions are far more complex because they can involve real and
distinct roots, real and equal roots and complex conjugate roots. But the solution
algorithms that are built into Maple handle all these. Furthermore, secondorder
differential equations involve two unknowns, which are denoted C1 and C2 in
Maple’s output.
The input instructions for some examples used in this chapter are shown in
table 2.7.
Initial value problems follow the same structure as before (table 2.8). 75 ...
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.
 Fall '11
 Dr.Gwartney
 Economics

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