By Brannan J., Boyce W.
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Extra info for Differential equations. An introduction to modern methods and applications
Therefore, by taking the exponential of both sides of Eq. (5), we find that |u − T0 | = e−kt+C = eC e−kt , (6) u − T0 = ±eC e−kt . (7) u = T0 + ce−kt (8) or Thus is a solution of Eq. (2), where c = ± eC is also an arbitrary (nonzero) constant. Note that if we allow c to take the value zero, then the constant solution u = T 0 is also contained in the expression (8). The sequence of steps leading from Eq. (2) to Eq. (8) proves that u = T 0 + ce−kt , where c is an arbitrary real number, is a solution of Eq.
C) Describe the behavior of the solution corresponding to the initial value a0 . y(1) = a, t > 0 24. t y + (t + 1)y = 2te−t , 25. t y + 2y = (sin t)/t, y(−π/2) = a, t < 0 26. (sin t)y + (cos t)y = et , y(1) = a, 0 < t < π 27. Consider the initial value problem y + 1 2 y = 2 cos t, y(0) = −1. Find the coordinates of the first local maximum point of the solution for t > 0. CONFIRMING PAGES 21:53 P1: KUF/OVY JWCL373-01 P2: OSO/OVY QC: SCF/OVY T1: SCF JWCL373-Brannan-v1 26 Chapter 1 October 12, 2010 Introduction 28.
32. Show that all solutions of 2y + t y = 2 [Eq. (36) of the text] approach a limit as t → ∞, and find the limiting value. Hint: Consider the general solution, Eq. (42), and use L’Hospital’s rule on the first term. 33. Show that if a and λ are positive constants, and b is any real number, then every solution of the equation y + ay = be−λt has the property that y → 0 as t → ∞. Hint: Consider the cases a = λ and a = λ separately. In each of Problems 34 through 37 construct a first order linear differential equation whose solutions have the required behavior as t → ∞.