更完整的内文 \begin{flushleft} \Large{Strauss PDE Exercises 1.2.} \end{flushleft} \\ \begin{flushleft} \begin{itemize} \item[{\large 1.2.8.}] Let \left\{ \begin{array}{c} x' = ax + by & y' = bx - ay \end{array} \right. $\Rightarrow$ \left\{ \begin{array}{c} u_{x} = au_{x'} + bu_{y'} & u_{y} = bu_{x'} - au_{y'} \end{array} \right. .\\ Therefore, \begin{align} & \text{ }au_{x}+bu_{y}+cu = a^{2} u_{x'} + b^{2} u_{x'}+cu =0 \notag \\ \Leftrightarrow & \text{ }u_{x'} = \frac{-c}{a^{2}+b^{2}} u \notag \\ \Leftrightarrow & \text{ }u = C_{1}(y') \exp \left(\frac{-c}{a^{2}+b^{2}} x' \right) =: f(bx-ay) \exp \left(\frac{-c(ax+by)}{a^{2}+b^{2}} \right) . \notag \end{align} \\ \end{itemize} \end{flushleft} \end{document} --

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