Economics Dynamics Problems 80

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

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.

Ask a homework question - tutors are online