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

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Answer

$x^{m} (m^{2} + 2 m) = \frac{\ln (\ln x)}{x}$ , 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: $m (m - 1) x^{m} + 3 m x^{m} + x^{m} = \frac{\ln (\ln x)}{x}$

Step 3 ： $x^{m} (m^{2} + 2 m) = \frac{\ln (\ln x)}{x}$ , 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.x2d2ydx2+3xdydx+y=ln⁡(ln⁡x)x

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