Question

\[\left( x - 1 \right)\frac{dy}{dx} = 2 xy\]

Answer 1

We have,
\[\left( x - 1 \right)\frac{dy}{dx} = 2 xy\]
\[ \Rightarrow \left( x - 1 \right)dy = 2xy dx\]
\[ \Rightarrow \frac{2x}{\left( x - 1 \right)}dx = \frac{1}{y}dy\]
Integrating both sides, we get
\[2\int\frac{x}{\left( x - 1 \right)}dx = \int\frac{1}{y}dy\]
\[ \Rightarrow 2\int\frac{x - 1 + 1}{x - 1}dx = \int\frac{1}{y}dy\]
\[ \Rightarrow 2\int dx + 2\int\frac{1}{x - 1}dx = \int\frac{1}{y}dy\]
\[ \Rightarrow 2x + 2 \log\left| x - 1 \right| = \log\left| y \right| + C\]