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Unformatted text preview: Show that if y is a solution and y(a) = 0 at some number a ≥ 0, then y(x) = 0 for all x. (A solution y is either identically zero or never zero.) ∗ 8.8 DIFFERENTIAL EQUATIONS; FIRST-ORDER LINEAR EQUATIONS 503 (b) Show that if r < 0, then all nonzero solutions are unbounded. (c) Show that if r > 0, then all solutions have limit 0 as x → ∞. (d) Describe all solutions of the equation in the case r = 0. Exercises 31 and 32 are concerned with the ﬁrst-order linear differential equation (∗) y + p(x)y = 0 Here we are measuring distance in feet and the positive direction is down. (a) Find v(t ). (b) Show that v(t ) cannot exceed 32/k and that v(t ) → 32/k as t → ∞. (c) Sketch the graph of v(t ). 38. Suppose that a certain population P has a known birth rate d B/dt and a known death rate dD/dt . Then the rate of change of P is given by dB dD dP = − . dt dt dt (a) Assume that dB/dt = aP and dD/dt = bP , where a and b are constants. Find P (t ) if P (0) = P0 > 0. (b) Analyze the cases (...
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- Spring '10