Economics Dynamics Problems 80

# Economics Dynamics Problems 80 - 64 Economic Dynamics The...

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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 Second-order linear nonhomogeneous equations A second-order 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.

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