\documentclass[a4paper,10pt]{article}
\begin{document}
{\bf HW1}
\begin{enumerate}


\item 
Derive the Euler equation. Solve
$$
\min_{u(x), b} \int_0^b \left( (u')^2 + 2 x \,u \right) dx  ~~ u(0) =0, ~~ u(b)= 0,
$$

 \item 


Derive the Euler equation and boundary conditions. Solve. Find the first integral
$$
\min_{u(x)}\left[ \int_0^1 \left( (u')^2 - \omega^2 u^2 \right) dx + 2
  u(1),\right] ~~ u(0) =0, 
$$
  

 \item 
Derive the Euler equation and boundary conditions. Solve. Explain the dependence of the solution on $a$.
$$
\min_{u(x)} \int_0^b \left( (u')^2 + a\, u\, u' + u\right) dx  ~~ u(0) =0.
$$


 \item 
Derive the Euler equation and boundary conditions. Solve, using Maple
$$
\min_{u(x)} \int_0^b \left( (u'')^2 - a\, (u')^2 + b\,u\right) dx  ~~ u(0)
=0,~~ u'(0)=0
$$

\end{enumerate}


\end{document}
