31 a show that if y1 and y2 are solutions of then u

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Unformatted text preview: i) a > b, (ii) a = b, (iii) a < b. Find lim P (t ) in each case. t →∞ where the function p is continuous on an interval I . 31. (a) Show that if y1 and y2 are solutions of (∗), then u = y1 + y2 is also a solution of (∗). (b) Show that if y is a solution of (∗) and C is a constant, then u = Cy is also a solution of (∗). 32. (a) Let a ∈ I . Show that the general solution of (∗) can be written as y(x) = Ce− x a p(t ) dt . (b) Show that if y is a solution of (∗) and y(b) = 0 for some b ∈ I , then y(x) = 0 for all x ∈ I . (c) Show that if y1 and y2 are solutions of (∗) and y1 (b) = y2 (b) for some b ∈ I , then y1 (x) = y2 (x) for all x ∈ I . Exercises 33 and 34 are concerned with the differential equation (8.8.1): y + p(x)y = q(x) where p and q are functions continuous on some interval I . 33. Let a ∈ I and let H (x) = x a 39. The current i in an electrical circuit consisting of a resistance R, inductance L, and voltage E varies with time (measured in seconds) accor...
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