29.10. EXERCISES 573
where r is the distance to the surface of the earth and here v = v(t) the speed of theprojectile at time t when it is at a distance of r from the surface of the earth. Nextexplain why
vdvdr
=− R2g
(R+ r)2
The two variables are v and r. Separate the variables and find the solution to thisdifferential equation given that the initial speed is v0 as stated above. Show that themaximum distance from the surface of the earth is given by
R
(Rg
Rg− 12 v2
0−1
)
provided that Rg > 12 v2
0. What is the smallest value of v0 such that the projectile willleave the earth and never return?
29. The Grompertz equation is dydt = ry ln
(Ky
). Find the solutions to this equation with
initial condition y(0) = y0. Also identify all equilibrium solutions and their stability.Also verify the inequality ry ln
(Ky
)≥ ry
(1− y
K
)for y ∈ [0,K]. Explain why for a
given initial condition y0 ∈ (0,K) , the solution to the Grompertz equation should beat least as large as the solution to the logistic equation.
30. You have a population which satisfies the logistic equation y′ = ry(1− y
K
)and the
initial condition is y(0) = αK where 0 < α < 1/2. How long will it take for thepopulation to double?
31. An equilibrium point is called semi-stable if it is stable from one side and not stablefrom the other. Sketch the appearance of f (y) near y0 if y0 is a semi-stable equilib-rium point. Here f (y0) = 0 and the differential equation is y′ = f (y).
32. Consider the differential equation y′ = a− y2 where a is a real number. Show thatthere are no equilibrium solutions if a < 0 but there are two of them if a > 0 and onlyone if a = 0. Discuss the stability of the two equilibrium points when a > 0. Whatabout stability of equilibrium when a = 0?
33. Do exactly the same problem when y′ = ay− y3. This time show there are threeequilibrium points when a > 0 and only one if a < 0. Discuss the stability of thesepoints.
34. Do the same problem if y′ = ay− y2. These three problems illustrate somethingcalled bifurcation which is when the nature of the solutions changes dramaticallywhen some parameter changes.
Homogeneous Equations
35. Find the solution curve to the following differential equations which contains thegiven point.
(a) y′ = 1x(2x+y) (x+ y)2 , (1,1)
(b) y′ =− 1x(x−2y)
(x2− xy+2y2
),(2,0)