Question

how to solve x²y''-3xy'-2y=0

Answer 1

$$x^2y''-3xy'-2y=0$$

$$y=x^m , y'=mx^{m-1}, y''=m(m-1)x^{m-2}$$

$$x^2(m(m-1)x^{m-2})-3x(mx^{m-1})-2x^m=0$$

$$(m^2-m-3m-2)x^m=0$$

$$(m^2-4m-2)x^m=0$$

$$m^2-4m-2=0$$

$$m=2\pm\sqrt{6}$$

$$y(x)=C_1x^{2-\sqrt{6}}+C_2x^{2+\sqrt{6}}$$