Practice Exam2

# Practice Exam2 - v if v x t = u x t-g x What are the...

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Math 445 - Spring 2008 Exam 2 Practice Page 1 1. Using separation of variables, solve: 2 u ∂t 2 = 4 2 u ∂x 2 with initial data u ( x, 0) = sin 3 x - 2 sin 5 x ∂u ∂t ( x, 0) = 0 and with Dirichlet boundary conditions u (0 , t ) = u ( π, t ) = 0 t > 0 2. Find all solutions u ( x ; , y ) = f ( x ) g ( y ) of the Laplace equation: 2 u ∂x 2 + 2 u ∂y 2 = 0 on the rectangle 0 x π , 0 y 2 π , where u is 0 on the three sides x = 0, y = 0, and y = 2 π . 3. Consider the equation: u t = u xx - 6 x - 2 with the homogeneous Dirichlet boundary conditions u (0 , t ) = 0 u (1 , t ) = 0 (a) First, find a function g ( x ) such that g ( x ) = 6 x + 2 and g (0) = g (1) = 0 (b) Write the equation satisfied by
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Unformatted text preview: v if v ( x, t ) = u ( x, t )-g ( x ). What are the boundary conditions for v ? (c) Solve the equation for v ( x, t ) using separation of variables if the initial condition for u is u ( x, 0) = x 3 + x 2 . 4. Find the type, transform to normal form and solve the PDE. u xy-u xx = 0 5. Solve the PDE for u ( x, t ) using fourier transform in space: u t = cu x ,-∞ < x < ∞ u ( x, 0) = f ( x ) Write the solution as a double integral and in terms of f via inverse fourier transform....
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