x^2*y''+2*x*y'-2*y=0

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2 2 d d -2*y(x) + x *---(y(x)) + 2*x*--(y(x)) = 0 2 dx dx

$$x^{2} \frac{d^{2}}{d x^{2}} y{\left(x \right)} + 2 x \frac{d}{d x} y{\left(x \right)} - 2 y{\left(x \right)} = 0$$

x^2*y'' + 2*x*y' - 2*y = 0

The answer (#2)
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$$y\left(x\right)={\it ilt}\left({{g_{19164}^2\,\left({{d^2}\over{d\,
g_{19164}^2}}\,\mathcal{L}\left(y\left(x\right) , x , g_{19164}
\right)\right)+2\,g_{19164}\,\left({{d}\over{d\,g_{19164}}}\,
\mathcal{L}\left(y\left(x\right) , x , g_{19164}\right)\right)
}\over{2}} , g_{19164} , x\right)$$

y = 'ilt((g19164^2*'diff('laplace(y,x,g19164),g19164,2)+2*g19164*'diff('laplace(y,x,g19164),g19164,1))/2,g19164,x)

The classification

2nd linear bessel

2nd power series regular

factorable

nth linear euler eq homogeneous