Consider the PDE: $$U_x+2U_y+(2x-y)U=2x^2+3xy-y^2 $$ in my lectures, we have found that using a change of coordiantes of the form: $$\bar{x}=ax+by,\bar{y}=bx-ay $$ can be useful for solving PDEs. However, we haven't been told how we would pick $a$ or $b$. The intention is for the PDE to boil down to an ODE. Is there a method for selecting $a$ and $b$?
2026-03-31 10:08:40.1774951720
How do we find a good change of coordiantes when solving PDEs?
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