70 Economic Dynamics We shall return to linear approximations in chapter 3 when considering differ-ence equations, and then again in chapters 4 and 5 when we deal with nonlinear systems of differential and difference equations. These investigations will allow us to use linear approximation methods when we consider economic models in part II. 2.11 Solving differential equations with Mathematica 2.11.1 First-order equations Mathematica has two built in commands for dealing with differential equations, which are the DSolve command and the NDSolve command. The Frst is used to Fnd a symbolic solution to a differential equation; the second Fnds a numerical approximation. Consider the following Frst-order differential equation dy dt = f ( y , t ) In particular, we are assuming that y is a function of t , y ( t ). Then we employ the DSolve command by using DSolve[y’[t]==f[y[t],t],y[t],t] Note a number of aspects of this instruction: (1) The equation utilises the single apostrophe, so y ± ( t
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