$\newcommand{\const}{\mathrm{const}}$ $\newcommand{\erf}{\operatorname{erf}}$
Deadline Thursday, April 2
It is 10-th Home assignment. Still only 7 best count
Consider variational problem under constrain: \begin{align} &\Phi(u):=\frac{1}{2}\int_0^2 u_{xx}^2\,dx,\label{eq-10.1}\\ &\Psi(u)= \frac{1}{2}\int_0^2 u_{x}^2\,dx =1,\label{eq-10.2}\\ &u(0)=u'(0)=u(2)=u'(2)=0.\label{eq-10.3} \end{align}
Hint: a. it will be 4th order ODE.
Hint: b. Changing interval to $(-1,1)$ and observing that ODE and boundary conditions are symmetric (survive $x\mapsto -x$) one can consider separately even and odd eigenfunctions and corresponding eigenvalues.
Consider variational problem \begin{equation} \Phi(u):=\frac{1}{2}\int_\Omega (u_x^2 + y u_y^2- 2 yu)\,dxdy \label{eq-10.4} \end{equation} where $\Omega=\{ (x,y):\, 0 < x< 1, 0 < y < 1\}$.
Consider distributions now. Calculate right from the definition
The results may be written through $\delta$-function or its derivatives of the smallest possible order.
Find Fourier transform of $\theta(x)$ (Heaviside function).
Consider wave equation \begin{equation} \rho u_{tt} - (k(x)u_x)_x=0 \label{eq-10.5} \end{equation} on $-\infty < x < \infty$ where $\rho(x)=\left\{\begin{aligned}\rho_-(x)& &&x<0,\\ \rho_+(x)& &&x>0\end{aligned}\right.$ and $k(x)=\left\{\begin{aligned}k_-(x)& &&x<0,\\ k_+(x)& &&x>0\end{aligned}\right.$.