Question

xy"+2y'=0 reduced into y

Answer 1

Let us solve $xy''+2y'=0.$ Let $z=y'.$ Then $xz'+2z=0.$ It follows that $x\frac{dz}{dx}=-2z,$ and hence $\frac{dz}{z}=-2\frac{dx}{x}.$ We have that $\int\frac{dz}{z}=-2\int\frac{dx}{x},$ and thus $\ln|z|=-2\ln|x|+\ln|C|=\ln|\frac{C}{x^2}|$ . We conclude that $z=\frac{C}{x^2}.$ Then $y'=\frac{C}{x^2},$ and hence the solution is $y=-\frac{C}{x}+C_2.$