Economics Dynamics Problems 96

Economics Dynamics Problems 96 - 80 Economic Dynamics Note...

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Unformatted text preview: 80 Economic Dynamics Note that the direction field has six elements: (i) (ii) (iii) (iv) (v) (vi) the differential equation y(x) indicates that x is the independent variable and y the dependent variable the range for the x-axis the initial points the range for the y-axis a set of options; here we have two options: (a) arrows are to be drawn slim (the default is thin) (b) the colour of the lines is to be blue (the default is yellow). Figure 2.9(a) (p. 46) This follows similar steps as figure 2.8 and so we shall be brief. We assume a new session. Input the following: (1) (2) (3) (4) (5) (6) with(DEtools): equ:=p0*exp(k*t); newequ:=subs(p0=13,k=0.01,equ); inisol:=evalf(subs(t=0,newequ)); finsol:=evalf(subs(t=150,newequ)); DEplot(diff(p(t),t)=0.01*p,p(t),t=0..150,{[0,13]}, p=0..60, arrows=slim,linecolour=blue); Instructions (2), (3), (4) and (5) input the equation and evaluate it for the initial point (time t = 0) and at t = 150. The remaining instruction plots the direction field and one integral curve through the point (0, 13). Figure 2.9(b) (p. 46) The logistic equation uses the values a = 0.02 and b = 0.000436 and p0 = 13. The input instructions are the following, where again we assume a new session: (1) (2) with(DEtools): DEplot(diff(p(t),t)=(0.02-0.000436*p)*p,p(t), t=0..150,{[0,13]},p=0..50,arrows=slim, linecolour=blue); Exercises 1. Show the following are solutions to their respective differential equations (i) dy = ky dx (ii) −x dy = dx y (iii) dy −2y = dx x y = cekx y = x 2 + y2 = c y= a x2 ...
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