Question

a) Consider the Cauchy problem for the linear one-dimensional wave equation \left\{\begin{array}{ll}

u_{u t}=u_{x x} & \text { for } x \in \mathbb{R} \text { and } t>0 \\

u(x, 0)=f(x) & \text { for } x \in \mathbb{R} \\

u_{t}(x, 0)=g(x) & \text { for } x \in \mathbb{R}

\end{array}\right. \text { where } f \in C^{2}(\mathbb{R}) \text { and } g \in C^{1}(\mathbb{R}) \text {. Show that if } f \text { and } g \text { are odd functions, } then for every fixed t > 0, the function (0,t) is necessarily equal to 0. (i) Without proving, write down the Laplace equation in polar coordi-nates and the formula for the general solution of the Laplace equation-in a disk in R² centred at (0,0) and of radius √/6. (ii) Let D = {(x,y) € R² = r² + y² <6}. Find a harmonic function inthe disk D, satisfying u(1, y) = y + y² on the boundary of D. Write-your answer in a Cartesian coordinate system.

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