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**Unformatted text preview: **MATH 412: Sample Questions for Midterm 2 1. Say whether each of the following functions f ( x ) is even or odd - or neither. If neither, write f ( x ) = f e + f o , where f e ( x ) is even and f o ( x ) is odd. (a) f ( x ) = x 5 sin( x 2 ) (b) f ( x ) = exp( x 2 ) (c) f ( x ) = 5 x + x 4 (d) f ( x ) = e x log | x | 2. Given the general statement of Fourier decomposition (the representation of a function as an infinite sum of orthogonal functions) ( x ) = X A n X n ( x ) derive an abstract expression for A k using the inner product for this problem. 3. Find the general solution by separation of variables to the following unusual diffusion equation between 0 and L , with Dirichlet boundary conditions: u t = Ktu xx , &lt; x &lt; L, t &gt; u (0 ,t ) = 0 , t &gt; u ( L,t ) = 0 , t &gt; , Note that there is an extra time dependence in the PDE, and that no initial conditions are given. 4. Solve the following problem: u tt = c...

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