The differential equation $\frac{dy}{dx} = \frac{y}{x} - \frac{1}{y}\;$ I am learning differential equations and can do the basic examples. However, how can you solve the differential equation
$$\frac{dy}{dx} = \frac{y}{x} - \frac{1}{y}\;?$$

$$\frac { dy }{ dx } =\frac { y }{ x } -\frac { 1 }{ y } \;$$ substitute

$$y=xt$$

$$  \frac { dy }{ dx } =t+x\frac { dt }{ dx } \\ t+x\frac { dt }{ dx } =t-\frac { 1 }{ xt } \\ x\frac { dt }{ dx } =-\frac { 1 }{ xt } \\ \int { tdt } =\int { -\frac { d }{ { x }^{ 2 } }  } \\ \frac { { t }^{ 2 } }{ 2 } =\frac { 1 }{ x } +C\\ \frac { { y }^{ 2 } }{ { x }^{ 2 } } =\frac { 2 }{ x } +C\\ y=\pm \sqrt { 2x+{ x }^{ 2 }C } $$
