Question

xy (y + 1) dy = (x² + 1) dx

Answer 1

We have, 
\[xy\left( y + 1 \right)dy = \left( x^2 + 1 \right)dx\]
\[ \Rightarrow \left\{ y\left( y + 1 \right) \right\}dy = \frac{x^2 + 1}{x}dx\]
\[ \Rightarrow \left( y^2 + y \right)dy = \left( x + \frac{1}{x} \right)dx\]
Integrating both sides, we get 
\[\int\left( y^2 + y \right)dy = \int\left( x + \frac{1}{x} \right)dx\]
\[ \Rightarrow \int y^2 dy + \int y dy = \int x dx + \int\frac{1}{x}dx\]
\[ \Rightarrow \frac{y^3}{3} + \frac{y^2}{2} = \frac{x^2}{2} + \log \left| x \right| + C\]
\[\text{ Hence,} \frac{y^3}{3} + \frac{y^2}{2} = \frac{x^2}{2} + \log \left| x \right| +\text{ C is the required solution }. \]