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Consider the Euler equation $x^2y''+4xy'+2y=0$, $x\in(1,2)$.

Given that it has solutions of the type $x^r$, the full list of solutions is:

$y=C_1e^{-x}+C_2e^{-2x}$

$y=C_1x^{-2+\sqrt{2}}+C_2x^{-2-\sqrt{2}}$

$y=C_1x^{-1}+C_2x^{-2}$

$y=C_1e^{(-2+\sqrt{2})x}+C_2e^{(-2-\sqrt{2})x}$