Economics Dynamics Problems 87

Economics Dynamics Problems 87 - often necessary It will be...

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Continuous dynamic systems 71 Table 2.1 First-order differential equations with Mathematica Problem Input instructions (i) dx dt = kx DSolve[x’[t]==kx[t],x[t],t] (ii) dx dt = 1 + ce t DSolve[x’[t]==1+cExp[t],x[t],t] (iii) dp dt α ( b d ) p = α ( a c ) DSolve[p’[t]- α (b-d)p[t]== α (a-c),p[t],t] (iv) dx dt = kx ( a x ) DSolve[x’[t]==kx[t](a-x[t]),x[t],t] (v) ˙ k = sak α ( n + δ ) k DSolve[k’[t]==sak[t] α -(n+ δ )k[t],k[t],t] Table 2.2 Mathematica input instructions for initial value problems Problem Input instructions (i) dp dt = p ( a bp ) , p (0) = p 0 DSolve[{p’[t]==p[t](a-bp[t]),p[0]==p0}, p[t],t] (ii) dn dt =− λ n , n (0) = n 0 DSolve[{n’[t]==- λ n[t],n[0]==n0},n[t],t] (iii) dy dx = x 2 2 x + 1 , y (0) = 1 DSolve[{y’[x]==x 2 -2x+1,y[0]==1},y[x],x] a way useful for economic interpretation. So some manipulation of the output is
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Unformatted text preview: often necessary. It will be noted that none of the above examples involve initial conditions, which is why all outputs involve the unknown constant C[1]. Initial value problems are treated in a similar manner. If we have the initial value problem, dy dt = f ( y , t ) y (0) = y then the input instruction is DSolve[{y’[t]==f[y[t],t],y[0]==y0},y[t],t] For example, look at table 2.2. 2.11.2 Second-order equations Second-order differential equations are treated in fundamentally the same way. If we have the homogeneous second-order differential equation a d 2 y dt 2 + b dy dt + cy =...
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