Economics Dynamics Problems 91

Economics Dynamics Problems 91 - Continuous dynamic systems...

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Unformatted text preview: Continuous dynamic systems Table 2.6 Maple input instructions for first-order 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)*(a-b*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^2-2*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 Second-order equations Second-order differential equations are treated in fundamentally the same way. If we have the homogeneous second-order 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 second-order 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, second-order 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.

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