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Economics Dynamics Problems 88

# Economics Dynamics Problems 88 - algorithms that are built...

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72 Economic Dynamics Table 2.3 Mathematica input instructions for second-order differential equations Problem Input Instructions (i) d 2 y dt 2 + 4 dy dt 5 y = 0 DSolve[y’’[t]+4y’[t]-5y[t]==0,y[t],t] (ii) y ( t ) + 4 y ( t ) + 4 y ( t ) = 0 DSolve[y’’[t]+4y’[t]+4y[t]==0,y[t],t] (iii) y ( t ) + 2 y ( t ) + 2 y ( t ) = 0 DSolve[y’’[t]+2y’[t]+2y[t]==0,y[t],t] (iv) y ( t ) + y ( t ) = t DSolve[y’’[t]+y’[t]==t,y[t],t] Table 2.4 Mathematica input instructions for initial value problems Problem Input instructions (i) d 2 y dt 2 + 4 dy dt 5 y = 0 , DSolve[ { y’’[t]+4y’[t]-5y[t]==0,y[0]==0, y (0) = 0 , y (0) = 1 y’[0]==1 } ,y[t],t] (ii) y ( t ) + 4 y ( t ) + 4 y ( t ) = 0 , DSolve[ { y’’[t]+4y’[t]+4y[t]==0,y[0]==3, y (0) = 3 , y (0) = 7 y’[0]==7 } ,y[t],t] then the input instruction is 11 DSolve[ay’’[t]+by’[t]+cy[t]==0,y[t],t] If we have the nonhomogeneous second-order differential equation a d 2 y dt 2 + b dy dt + cy = g ( t ) then the input instruction is DSolve[ay’’[t]+by’[t]+cy[t]==g[t],y[t],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
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Unformatted text preview: algorithms that are built into Mathematica handle all these. Furthermore, second-order differential equations involve two unknowns, which are denoted C[1] and C[2] in Mathematica ’s output. The Mathematica input instructions for some examples used in this chapter are shown in table 2.3. Initial value problems follow the same structure as before (table 2.4). 2.11.3 NDSolve Many differential equations, especially nonlinear and nonautonomous differential equations, cannot be solved by any of the known solution methods. In such cases a numerical approximation can be provided using the NDSolve command. In using NDSolve it is necessary, however, to provide initial conditions and the range for 11 Do not use the double quotes in these equations; rather input the single quote twice....
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