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s11_mthsc208_hw19

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

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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 = cosh 2 t + sinh 2 t and e - 2 t = cosh 2 t - sinh 2 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 π and t 0, which satisfies the following conditions: ∂u ∂t = c 2 2 u ∂x 2 , u (0 , t ) = u ( π, t ) = 0 , u ( x, 0) = 5 sin x + 3 sin 2 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 = π ) from the boundary conditions for u ( x, t ).

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s11_mthsc208_hw19 - MthSc 208 Fall 2011(Differential...

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