4.1. Find the general solution of the ODE $y^{\prime}=\frac{y^{2}}{x}$.
Solution. Rewrite the equation as $\frac{d y}{d x}=\frac{y^{2}}{x}$. Gather all terms depending on $y$ on the left side, and all terms depending on $x$ on the right side: $\frac{d y}{y^{2}}=\frac{d x}{x}$. Integrate both sides, each with respect to its own variable: $\int \frac{d y}{y^{2}}=\int \frac{d x}{x}$; it is convenient for further obtaining the solution in explicit form to represent the arbitrary constant of integration as $\ln C:-\frac{1}{y}=\ln x+\ln C-$ we have obtained the general solution of the ODE in implicit form. In explicit form, the solution is $y=-\frac{1}{\ln (C x)}$.