? 1 the phase diagram is shown in figure 137 if the

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1 will move outwards at a constant rate. λ = 1. The phase diagram is shown in Figure 1.37. If the bead is placed at rest at any point on the wire then it will remain in that position subsequently.
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1 : Second-order differential equations in the phase plane 29 2 1 1 2 x 2 1 1 2 y Figure 1.37 Problem 1.19: Phase diagram for λ = 0. 1.20 A particle is attached to a fixed point O on a smooth horizontal plane by an elastic string. When unstretched, the length of the string is 2 a . The equation of motion of the particle, which is constrained to move on a straight line through O , is ¨ x = − x + a sgn (x) , | x | > a (when the string is stretched), ¨ x = 0, | x | ≤ a (when the string is slack), x being the displacement from O . Find the equilibrium points and the equations of the phase paths, and sketch the phase diagram. 1.20. The equation of motion of the particle is ¨ x = − x + a sgn (x) , ( | x | > a) ¨ x = 0, ( | x | ≤ a) . All points in the interval | x | ≤ a , y = 0 are equilibrium points. The phase paths as follows. (i) x > a . The differential equation is d y d x = x + a y , which has the general solution y 2 + (x a) 2 = C 1 . These phase paths are semicircles centred at (a , 0 ) .
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30 Nonlinear ordinary differential equations: problems and solutions x y a a Figure 1.38 Problem 1.20. (ii) a x a . The phase paths are the straight lines y = C 2 . (iii) x < a . The differential equation is d y d x = x a y , which has the general solution y 2 + (x + a) 2 = C 3 . These phase paths are semicircles centred at ( a , 0 ) . A sketch of the phase paths is shown in Figure 1.38. All paths are closed which means that all solutions are periodic. 1.21 The equation of motion of a conservative system is ¨ x + g(x) = 0, where g( 0 ) = 0, and g(x) is strictly increasing for all x , and x 0 g(u) d u → ∞ as x → ±∞ . (i) Show that the motion is always periodic. By considering g(x) = x e x 2 , show that if (i) does not hold, the motions are not all necessarily periodic. 1.21. The equation for the phase paths is d y d x = − g(x) y . The variables separate to give the general solution in the form 1 2 y 2 = − x 0 g(u) d u + C . ( i )
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1 : Second-order differential equations in the phase plane 31 Write x 0 g(u) d u = G(x) . ( ii ) Then (i) defines two families of paths where C > 0; y = 2 { C G(x) } 1 / 2 when G(x) < C ; ( iii ) and the reflection in the x axis; y = − 2 { C G(x) } 1 / 2 when G(x) < C . ( iv ) Since g(x) < 0 when x < 0, and g(x) > 0 when x > 0, then G(x) is strictly increasing to +∞ as x → −∞ and x → ∞ . Also G(x) is continuous and G( 0 ) = 0. Therefore, given any value of C > 0, G(x) takes the value C at exactly two values of x , one negative and the other positive. Consider the family of positive solutions (iii). Take any positive value of the constant C . At the two points where G(x) = C , we have y(x) = 0. Between them y(x) > 0, and the graph of the path cuts the x axis at right angles (see Section 1.2). When the corresponding reflected curve (iv) (y < 0 ) is joined to this one, we have a smooth closed curve. By varying the parameter C the process generates a family of closed curves nested around the origin (which is therefore a centre), and all motions are periodic.
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