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Unformatted text preview: CAAM 336 Â· DIFFERENTIAL EQUATIONS Examination 2 Posted Monday, 8 December 2008. Due no later than 5pm on Wednesday, 17 December 2008. Instructions: 1. Time limit: 4 uninterrupted hours . 2. There are three questions worth a total of 100 points, plus a 10 point bonus. Please do not look at the questions until you begin the exam. 3. You may not use any outside resources, such as books, notes, problem sets, friends, calculators, or MATLAB. 4. Please answer the questions thoroughly and justify all your answers. Show all your work to maximize partial credit. 5. Print your name on the line below: 6. Time started: Time completed: 7. Indicate that this is your own individual effort in compliance with the instructions above and the honor system by writing out in full and signing the traditional pledge on the lines below. 8. Staple this page to the front of your exam. CAAM 336 Â· DIFFERENTIAL EQUATIONS 1. [50 points: 8 points each for (a), (e), (f), (g); 6 points each for (b), (c), (d)] We can apply the techniques learned in this course to solve a variety of linear partial differ- ential equations. In this problem, you will apply these techniques to solve the telegrapherâ€™s equation , a generalization of the damped wave equation from Problem Set 11 that arises as a model of a lossy electrical transmission line: w tt ( x,t ) = c 2 w xx ( x,t ) + bw ( x,t )- 2 Î´w t ( x,t ) with positive constants c , b , and Î´ . We impose homogeneous Dirichlet boundary conditions, w (0 ,t ) = w (1 ,t ) = 0. (a) Show that with the substitution w ( x,t ) = e- Î´t u ( x,t ), the telegrapherâ€™s equation can be reduced to the Kleinâ€“Gordon equation u tt ( x,t ) = c 2 u xx ( x,t ) + ( b + Î´ 2 ) u ( x,t ) (cf. Polyanin; the Kleinâ€“Gordon equation is of independent interest as a particle model in quantum mechanics)....
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- Fall '09
- Partial differential equation, wave equation