44Economic DynamicsWe can summarise the method of isoclines as follows:(1)From the differential equationdydx=φ(x,y)determine the family of isoclinesφ(x,y)=mand construct several members of this family.(2)Consider a particular isoclineφ(x,y)=m0. All points (x,y) on this iso-cline have the same slopem0. Obtain the inclinationα0=arctanm00≤α0≤180◦Along the isoclineφ(x,y)=m0construct line elements with inclinationα0. (This establishes part of the direction field.)(3)Repeat step 2 for each isocline.(4)Draw smooth curves to represent the approximate integral curves indi-cated by the line elements of step 3.It is apparent that this is a very tedious procedure. Luckily, a number of mathe-matical software packages now compute direction fields and can be used to con-struct isoclines (see appendices 2.1 and 2.2).Example 2.12dydx=2x−yIn sections 2.11 and 2.12 we give the instructions on using software pack-ages to solve this differential equation explicitly, and in appendices 2.1 and2.2 we provideMathematicaandMapleinstructions, respectively, for plot-
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