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1. Use the method of characteristics to solve the boundary value problem du 0. = 3x x € R. 2. (a) Apply the method of separation of variables to the heat equation au F²u Dit 0r2 with Neumann boundary conditions for t > 0, and initial condition u(x,0)=f(x). where / [0, 1] - R is a given continuously differentiable function. (b) Find the limit lim a(x.1) 1-900 when a is a solution to the IBVP delined in the previous part when f(x) = cos²(2=x). 3. Find a solution to the IBVP aw aw x, (x.t) E (0.oc) x (0, xo). Or of You may use any mathematical method you want excluding merely writing a solution and verifying that it is a solution. 4. (a) At what points. if any, is the function / C - C defined by f(2)=2E. analytic? (b) Find the value of the line integral of 16(2+ d2, where C is the circle |z| = 10.000,000 parameterized counterclockwise. (Hint: You may use without proof the following derivatives: d(e") d(sin 2) d(cos 2) = cost = sin 2 d2 d2 d2 and the product rule.)

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Differential Equations
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