Solve the Cauchy-Euler equation.
\[
x^{2} \frac{d^{2} y}{d x^{2}}+3 x \frac{d y}{d x}+y=\frac{\ln (\ln x)}{x}
\]

Answer

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Answer

\(x^m(m^2 + 2m) = \frac{\ln(\ln x)}{x}\), but x^m can't satisfy the right side and the given equation is a nonhomogeneous Cauchy-Euler equation, which can't be solved directly.

Steps

Step 1 ：Let y(x) = x^m, then \(y'(x) = m x^{m-1}\) and \(y''(x) = m(m-1)x^{m-2}\)

Step 2 ：Substitute y(x), y'(x), and y''(x) into the given equation: \(m(m-1)x^{m} + 3mx^{m} + x^{m} = \frac{\ln(\ln x)}{x}\)

Step 3 ：\(x^m(m^2 + 2m) = \frac{\ln(\ln x)}{x}\), but x^m can't satisfy the right side and the given equation is a nonhomogeneous Cauchy-Euler equation, which can't be solved directly.

Solve the Cauchy-Euler equation.\[x^{2} \frac{d^{2} y}{d x^{2}}+3 x \frac{d y}{d x}+y=\frac{\ln (\ln x)}{x}\]

Answer

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Solve the Cauchy-Euler equation.
\[
x^{2} \frac{d^{2} y}{d x^{2}}+3 x \frac{d y}{d x}+y=\frac{\ln (\ln x)}{x}
\]