Question

how to solve xy''+y'=0

Answer 1

$$xy''+y'=0$$

$$y'=u, y''=u'$$

$$xu'+u=0$$

$$x\dfrac{du}{dx}=-u$$

$$\dfrac{du}{u}=-\dfrac{dx}{x}$$

$$\int\dfrac{du}{u}=-\int\dfrac{dx}{x}$$

$$\ln|u|=-\ln|x|+\ln C_1$$

$$u=\dfrac{C_1}{x}$$

$$y'=\dfrac{C_1}{x}$$

$$\int dy=\int\dfrac{C_1}{x}dx$$

$$y=C_1\ln |x|+C_2$$