The general solution to the Cauchy-Euler differential equation $x^2 \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + 2y = 0$ for $x > 0$ is given by:

(A) $y(x) = C_1 x + C_2 x^2$

(B) $y(x) = C_1 x^{-1} + C_2 x^{-2}$

(C) $y(x) = (C_1 + C_2 \ln x) x$

(D) $y(x) = C_1 \cos(\ln x) + C_2 \sin(\ln x)$

Answer Key:

A

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Equidimensional Form:** An equation of the form $x^2 y'' + a x y' + b y = 0$ has solutions of the form $y = x^m$. 2. **Stage 2: Characteristic Polynomial:** * $x^2 [m(m-1) x^{m-2}] - 2x [m x^{m-1}] + 2 [x^m] = 0$. * Dividing by $x^m \ne 0$: * $m(m-1) - 2m + 2 = 0 \implies m^2 - m - 2m + 2 = 0 \implies m^2 - 3m + 2 = 0$. 3. **Stage 3: Roots:** * $(m - 1)(m - 2) = 0 \implies m_1 = 1, m_2 = 2$. 4. **Stage 4: General Solution:** Since roots are real and distinct: $y(x) = C_1 x^1 + C_2 x^2 = C_1 x + C_2 x^2$. 5. **Stage 5: Official Reference:** Dennis G. Zill, *A First Course in Differential Equations*, Chapter 4.

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