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{\large{\bf HW 3}}


1. At a gas planet with variable viscosity, the travel speed is
proportional to the radius $r$ from the center, $v(r)= \alpha\, r$ . What is the fastest path from a point
$A={r_1, \theta_1 )$  to a point $B=(r_2, \theta_2) $, where $\theta$ is a
polar angle in the plane that passes through the center and points $A$
and $B$.

2.  Show that the problem
$$
\min_{u(x)} \int _{-\pi} ^\pi \left[(u')^2 (1-\cos x)\right] dx, \quad
u(-\pi)=-1, ~u(\pi)=1
$$
does not have a regular (continuous) solution. Regularize the problem, find the solution, and plot the graphs of the extremals with several values of the regularization parameter. 

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