For the Cauchy–Euler homogeneous linear differential equation $x^2 \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + 2y = 0$ for $x > 0$, the sum of the roots of the auxiliary characteristic equation is ________ (answer in integer).
Answer Key:
3
Step-by-Step Solution & Explanation
Substitute $x = e^z$ (so $z = \ln x$). The operators transform as:
$x \frac{d}{dx} = D, \quad x^2 \frac{d^2}{dx^2} = D(D-1)$
where $D = \frac{d}{dz}$.
Substituting into the differential equation:
$[D(D-1) - 2D + 2]y = 0$
$(D^2 - 3D + 2)y = 0$
The auxiliary equation is:
$m^2 - 3m + 2 = 0 \implies (m - 1)(m - 2) = 0$
The roots are $m_1 = 1$ and $m_2 = 2$.
The sum of the roots is $1 + 2 = 3$.