Unformatted text preview: 64 Economic Dynamics
The coefﬁcients are
c1 = y(0) = 2
c2 = y (0) − αy(0)
=3
β Hence the particular solution is
y(t) = 2e−t cos(t) + 3e−t sin(t) 2.9 Secondorder linear nonhomogeneous equations
A secondorder linear nonhomogeneous equation with constant coefﬁcients takes
the form
a (2.29) dy
d2 y
+ b + cy = g(t)
dt2
dt or
ay (t) + by (t) + cy(t) = g(t)
Let L(y) = ay (t) + by (t) + cy(t) then equation (2.29) can be expressed as L( y) =
g(t). The solution to equation (2.29) can be thought of in two parts. First, there
is the homogeneous component, L( y) = 0. As we demonstrated in the previous
section, if the roots are real and distinct then
yc = c1 ert + c2 est
The reason for denoting this solution as yc will become clear in a moment.
Second, it is possible to come up with a particular solution, denoted yp , which
satisﬁes L( yp ) = g(t). yc is referred to as the complementary solution satisfying
L(y) = 0, while yp is the particular solution satisfying L( yp ) = g(t). If both yc
and yp are solutions, then so is their sum, y = yc + yp , which is referred to as
the general solution to a linear nonhomogeneous differential equation. Hence,
the general solution to equation (2.29) if the roots are real and distinct takes the
form
y(t) = yc + yp = c1 ert + c2 est + yp
The general solution y(t) = yc + yp holds even when the roots are not real or
distinct. The point is that the complementary solution arises from the solution to
L(y) = 0. As in the previous section there are three possible cases:
(1) Real and distinct roots
yc = c1 ert + c2 est (2) Real and equal roots
yc = c1 ert + c2 tert ...
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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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