1. Use the method of characteristics to solve the boundary value problem
= 3x x € R.
2. (a) Apply the method of separation of variables to the heat equation
with Neumann boundary conditions
for t > 0,
and initial condition
where / [0, 1] - R is a given continuously differentiable function.
(b) Find the limit
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
x, (x.t) E (0.oc) x (0, xo).
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
(b) Find the value of the line integral of
where C is the circle |z| = 10.000,000 parameterized counterclockwise.
(Hint: You may use without proof the following derivatives:
and the product rule.)
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