Question

Solve the differential equation $x\frac{dy}{dx}-x^2=\frac{2}{3}\log x$.

Answer 1

$x\frac{dy}{dx}-x^2=\frac{2}{3}\log xx\frac{dy}{dx}=\frac{2}{3}\log x+x^2\frac{dy}{dx}=\frac{1}{x}(\frac{2}{3}\log x+x^2)\frac{dy}{dx}=\frac{2}{3}\frac{\log x}{x}+xy=\int \frac{2}{3}\frac{\log x}{x}+xdxy=\frac{2}{3}\int \frac{\log x}{x}dx+\int xdxy=\frac{2}{3}\frac{(\log x{)}^2}{2}+\frac{x^2}{2}y=\frac{(\log x{)}^2}{3}+\frac{x^2}{2}$