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8) Ids The scalar function - z(t) I thus satisfies the inequality ~(t) for every c > 0. :::; c + m Jo' ~(8) d8 On the other hand, (19) may be written < ~(t) c and, since ~(t) (19) + m J0' ~(s) ds- 1 is continuous, ! o' ~(8) ds) J< 1 This in turn implies that In [ c + m 1:' ~(s) d8] < ml < t < b, and hence for 0 c + m Jo' (s) ds < ce"" (20) Using (19) and (20), we have finally (t) (21) < t < b. Since (21) holds for every c > 0, necessarily = 0 or, what is the same, x(t) = z(t) for 0 < t < b, which for 0 ~(t) < ce"" was to be proved.

3 Consider the following system of two coupled second-order equations. d 2x dy dt" + 5 dt + X - 2y = 0 (51) 2 -ddty2 + 2dydt- - 3x + y = 0 Let x = Xt, dx/dt = x2, y = x 3, and dy/dt = Xt. Then the system (51) is equivalent to the vector equation (48) with 0 1 -1 0 A- ( 0 0 3 0 0 0) 2 0 -1 -5 1 -2 ExAMPLE 4 The linear differential equation d 2x dt 2 + a(t) dx dt + b(t) x = f(t) {52) is equivalent to the vector equation (53) where and provided 32 NONLINEAR DIFFERENTIAL EQUATIONS The vector equation (53) represents the most general linear system of dimension two.

9. The equation (ii) 24· NONLINEAR DIFFERENTIAL EQUATIONS where k is a small positive quantity, appears in the theory of equatorial satellite orbits of an oblate spheroid. ) Discuss the phase-plane trajectories of (ii), and express the solutions in terms of standard elliptic functions. 10. Discuss the phase-plane trajectories of the equation d 2x a dt 2 +X= b- X where each of a and b is a positive constant. Consider various possibilities in regard to the relative magnitudes of a and b. ) 11. Show that the phase-plane trajectories of the equation ~; + f(x) = 0 (iii) are given by y2 where F(x) = = c- F(x) (iv) 2/o z f(u) du and c is an integration constant.

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An Intro to the Study of the Elements of the Diff and Int Calculus by A. Harnack

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