This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 4 20. The auxiliary equation for this differential equation, r 2 4 r + 4 = ( r 2) 2 = 0, has a double root r = 2. Thus, two linearly independent solutions are y 1 ( t ) = e 2 t and y 2 ( t ) = te 2 t . This means that a general solution is given by y ( t ) = ( c 1 + c 2 t ) e 2 t . Substituting the initial conditions into the general solution and its derivative yields y (1) = ( c 1 + c 2 t ) e 2 t  t =1 = ( c 1 + c 2 ) e 2 = 1 y (1) = ( c 2 + 2 c 1 + 2 c 2 t ) e 2 t  t =1 = (2 c 1 + 3 c 2 ) e 2 = 1 . So, c 1 = 2 e 2 and c 2 = e 2 . Therefore, the solution is y ( t ) = ( 2 e 2 e 2 t ) e 2 t = (2 t ) e 2 t 2 . 22. We substitute y = e rt into the given equation and get 3 re rt 7 e rt = (3 r 7) e rt = 0 . Therefore, 3 r 7 = 0 ⇒ r = 7 3 , and a general solution to the given differential equation is y ( t ) = ce 7 t/ 3 , where c is an arbitrary constant. 24. Similarly to the previous problem, we find the characteristic equation, 3 r +11 = 0, which has the root r = 11 / 3. Therefore, a general solution is given by z ( t ) = ce 11 t/ 3 . 26. (a) Substituting boundary conditions into y ( t ) = c 1 cos t + c 2 sin t yields 2 = y (0) = c 1 0 = y ( π/ 2) = c 2 . Thus, c 1 and c 2 are determined uniquely, and so the given boundary value problem has a unique solution y = 2 cos t . (b) Similarly to part (a), we obtain a system to determine c 1 and c 2 . 2 = y (0) = c 1 0 = y ( π ) = c 1 . However, this system is inconsistent, and so there is no solution satisfying given boundary conditions. 106 Exercises 4.2 (c) This time, we come up with a system 2 = y (0) = c 1 2 = y ( π ) = c 1 , which has infinitely many solutions given by c 1 = 2 and c 2 – arbitrary. Thus, the boundary value problem has infinitely many solutions of the form...
View
Full
Document
This note was uploaded on 02/12/2011 for the course MA 221 taught by Professor Mazmani during the Spring '08 term at Stevens.
 Spring '08
 MAZMANI
 Equations

Click to edit the document details