Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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Solving the LaPlace PDE by complete induction

I have a problem with an induction step that appears to be bothersome for me to comprehend it. Problem: Show that $f(x_1, \dots ,x_n)=(x_1^2+ \dots + x_n^2)^{1- n/2}$ for $(x_1^2 + \dots + ...