s11_mthsc208_hw19

s11_mthsc208_hw19 - MthSc 208, Fall 2011 (Differential...

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Unformatted text preview: MthSc 208, Fall 2011 (Differential Equations) Dr. Macauley HW 19 Due Friday April 22nd, 2011 (1) Consider the ODE y 00 = 4 y . We know that the general solution is y ( t ) = C 1 e 2 t + C 2 e- 2 t , i.e., { e 2 t ,e- 2 t } is a basis for the solution space. Use the fact that e 2 t = cosh2 t + sinh2 t and e- 2 t = cosh2 t- sinh2 t , and that any linear combination of solutions is a solution, to find two distinct solutions involving hyperbolic sines and cosines. Write the general solution using these functions. (2) We will solve for the function u ( x,t ), defined for 0 x and t 0, which satisfies the following conditions: u t = c 2 2 u x 2 , u (0 ,t ) = u ( ,t ) = 0 , u ( x, 0) = 5sin x + 3sin2 x. (a) Briefly describe, and sketch, a physical situation which this models. Be sure to explain the effect of both boundary conditions (called Dirichlet boundary conditions) and the initial condition. (b) Assume that u ( x,t ) = f ( x ) g ( t ). Find u t and u xx . Also, determine the boundary conditions for f ( x ) (at x = 0 and x = ) from the boundary conditions for...
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This note was uploaded on 03/11/2012 for the course MTHSC 208 taught by Professor Staufeneger during the Spring '09 term at Clemson.

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s11_mthsc208_hw19 - MthSc 208, Fall 2011 (Differential...

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