For the Cauchy-Euler differential equation x^2 (d^2y/dx^2) - 2x (dy/dx) + 2y = 0 for x > 0, what is the general solution?
Answer and explanation
Correct answer: D. C1 x + C2 x^2
- Step 1: Substitute y = x^m: m(m - 1) - 2m + 2 = 0.
- Step 2: Expand: m^2 - m - 2m + 2 = m^2 - 3m + 2 = 0.
- Step 3: Factoring gives: (m - 1)(m - 2) = 0 => m = 1, 2.
- Step 4: The general solution is y = C1 x^1 + C2 x^2 = C1 x + C2 x^2.
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