Math 6420 • PDE

Find the Euler equations and boundary conditions for the
minimization problems of the functionals
\[
  I_1=\min_u\int_0^1\left((u'')^2+x(u')^2\right)\,dx,
\]
and
\[
  I_2=\min_u\int_0^1\left((u'')^2-uu''+(x-1)(u')^2\right)\,dx,
\]
if \(u(0)=1\) and \(u'(1)=2\) for both problems. Why do their Euler equations coincide?
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\examheading{Part C}

Do two of the four problems for full credit on part C.

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