Question

Using the method of characteristics, solve the equation u_{x}+x u_{y}=u \text { for } x>1, y \in \mathbb{R} \text {. } \text { subject to the initial condition } u(1, y)=\sin y \text { for } y \in \mathbb{R} \text {. } ) Consider the following second order linear PDE 4 y^{2} u_{x x}-e^{2 x} u_{V y}-4 y^{2} u_{x}=0 \text { for } x \in \mathbb{R} \text { and } y<0 Determine its type on the given domain. (ii) Determine a suitable change of variables, § = {(1, y) and n = n(x, y)which, after putting v(§, n) = u(x, y), gives the canonical form of the-transformed equation. Do not bring the equation to its canonical form.

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