I understand that for a basic equation of this form, we just want to make it an ODE so it is baby stuff to solve. Not really sure what to do with 3u since the derivative/chain rule don't apply. Thanks!
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If $\displaystyle u_s = u_{x_1} {x_1}_s + u_{x_2} {x_2}_s + u_{x_3} {x_3}_s$ and with your chose of $\displaystyle {x_1}_s = x_1,\;\; {x_2}_s = 2x_2,\;\; {x_3}_s = 1$ gives via your PDE $\displaystyle u_s = 3u$.
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