Section 7.5

Section 7.5 - Differential Equations Math 308, Spring 2008...

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Unformatted text preview: Differential Equations Math 308, Spring 2008 Texas A&M University c F. Dos Reis Section 7.5 Recall form the previous chapters: f (t) 1 eax tn n = 1, 2, sin bt cos bt eat tn n = 1, 2, eat sin bt eat cos bt F (s) 1 s 1 s-a n! sn+1 s2 s2 b + b2 s + b2 n! (s - a)n+1 b (s - a)2 + b2 s-a (s - a)2 + b2 L eat f (t) = L {f (t)} (s - a) L f (s) = sL {f } (s) - f (0). L f (s) = s2 L {f } (s) - f (0) - sf (0). L f (n) (s) = sn L {f } (s) - sn-1 f (0) - sn-2 f (0) - - f (n-1) (0). L {tn f (t)} (s) = (-1)n Exercise 1. Solve the initial value problem y + 6y + 5y = 12 et , using Laplace transforms. 1 y(0) = -1, y (0) = 7 dn F (s). dsn Differential Equations Math 308, Spring 2008 Texas A&M University c F. Dos Reis Method of Laplace transforms: To solve an initial value problem: 1. Take the Laplace transform of both sides of the equation. 2. Use the properties of the Laplace transfom and the initial conditions to obtain an equation for the Laplace transform of the solution and then solve this equations for the transform. 3. Determine the inverse Laplace transform of the solution. Exercise 2. (5p401) Solve the initial value problem w + w = t2 + 2, w(0) = 1, w (0) = -1 Exercise 3. (9p401) Solve the initial value problem z + 5z - 6z = 21 et-1 , z(10 = -1, z (1) = 9 Exercise 4. (35p402) Find a solution to the initial value problem y + 3ty - 6y = 1, y(0) = 0, y (0) = 0 Exercise 5. (27p 402) Solve the initial value problem y + 3y + 3y + y = 0, y(0) = -4, y (0) = 4, y (0) = -2 2 ...
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Section 7.5 - Differential Equations Math 308, Spring 2008...

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