s09_mthsc208_hw22 - MTHSC 208(Dierential Equations Dr...

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MTHSC 208 (Differential Equations) Dr. Matthew Macauley HW 22 Due Wednesday April 15th, 2009 (1) Let X be a vector space over C (i.e., the contants are complex numbers, instead of just real numbers). If { v 1 , v 2 } is a basis of X , then by definition, every vector v can be written uniquely as v = C 1 v 1 + C 2 v 2 . (a) Is the set { 1 2 v 1 + 1 2 v 2 , 1 2 v 1 - 1 2 v 2 } a basis of X ? (b) Consider the ODE y = 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 (b), and the fact that e x = cosh x + sinh x to find a basis for the solution space involving hyperbolic sines and cosines, and write the general solution using these functions. (2) We will find the function u ( x, t ), defined for 0 x π and t 0, which satisfies the following conditions: ∂u ∂t = 9 2 u ∂x 2 , u (0 , t ) = u ( π, t ) = 0 , u ( x, 0) = sin x + 3 sin 2 x - 5 sin 7 x. (a) Assume that u ( x, t ) = f ( x ) g ( t ). Plug this back into the PDE and separate variables to get the eigenvalue problem (set equal to a constant λ ). Solve for g ( t ), f ( x ), and
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