Differential Equations

For the Cauchy-Euler ODE $x^2 y'' - 2x y' + 2y = 0$ for $x > 0$, the auxiliary equation in $m$ is $m(m - 1) - 2m + 2 = m^2 - 3m + 2 = 0$, giving roots $m = 1, 2$. If $y(x) = C_1 x + C_2 x^2$ with $y(1) = 3$ and $y(2) = 8$, then $C_1 + C_2 = 3$ and $2 C_1 + 4 C_2 = 8 \implies C_1 = 2, C_2 = 1$. The value of $y(3)$ is: