Economics Dynamics Problems 93

Economics Dynamics Problems 93 - dy dx = 2 x − y the...

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Continuous dynamic systems 77 the instruction for solving this is Sol1=dsolve({diff(y(t),t)=sin(3*t-y(t)),y(0)=0.5}, y(t),numeric); Although the output is named ‘Sol’, the solution is still for the variable y ( t ). So the plot would involve the input odeplot(Sol1,[t,y(t)],0. .10); Note that the range for t is given only in the odeplot instruction. Higher-order ordinary differential equations are treated in the same way. For example, given the initial value problem, d 2 y dt 2 + 0 . 5 dy dt + sin( y ) = 0 , y (0) =− 1 , y ± (0) = 0 , t [0,15] the input instruction is Sol2=dsolve({diff(y(t),t$2)+0.5*diff(y(t), t)+sin(y(t))=0, y(0)=-1,D(y)(0)=0},y(t),numeric); with plot odeplot(Sol2,[t,y(t)],0. .15); Appendix2.1PlottingdirectionfeldsForasingleequation with Mathematica ±igure 2.8 (p. 45) Given the differential equation
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Unformatted text preview: dy dx = 2 x − y the direction ±eld and isoclines can be obtained using Mathematica as follows: Step 1 Load the Plot±ield subroutine with the instruction << Graphics`PlotField` Note the use of the back-sloped apostrophe. Step 2 Obtain the direction ±eld by using the PlotVector±ield command as follows arrows=PlotVectorField[{1,2x-y},{x,-2,2},{y,-2,2}] Note the following: (a) ‘arrows’ is a name (with lower case a) which will be used later in the routine (b) the ±rst element in the ±rst bracket is unity, which represents the time derivative with respect to itself (c) if memory is scarce, the plot can be suppressed by inserting a semi-colon at the end of the line....
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