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

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