## anonymous 5 years ago How do I find two linear solutions for the linear equation: y" + (1/x)y' = 0 ?

Hi hollyn, If you make a substitution$v=y'$then$v'=y''$and your equation becomes$v'+\frac{1}{x}v=0$This equation is separable,$\frac{dv}{dx}=-\frac{v}{x} \rightarrow \frac{dv}{v}=-\frac{dx}{x}$Integrate both sides$\ln v = - \ln x +c$where c is some constant.Exponentiating both sides leaves you with$v=\frac{c_1}{x}$where c_1 is some constant. But remember, $v=y'$so$\frac{dy}{dx}=\frac{c_1}{x} \rightarrow y=c_1 \ln x + c_2$where c_2 is an arbitrary constant.