Question

3x² dy = (3xy + y²) dx

Answer 1

We have, 
\[3 x^2 dy = \left( 3xy + y^2 \right) dx\]
\[ \Rightarrow \frac{dy}{dx} = \frac{3xy + y^2}{3 x^2}\]
This is a homogeneous differential equation .
\[\text{ Putting }y = vx\text{ and }\frac{dy}{dx} = v + x\frac{dv}{dx}, \text{ we get }\]
\[v + x\frac{dv}{dx} = \frac{3v x^2 + v^2 x^2}{3 x^2}\]
\[ \Rightarrow v + x\frac{dv}{dx} = \frac{3v + v^2}{3}\]
\[ \Rightarrow x\frac{dv}{dx} = \frac{v^2}{3}\]
\[ \Rightarrow \frac{3}{v^2}dv = \frac{1}{x}dx\]
Integrating both sides, we get
\[3\int\frac{1}{v^2}dv = \int\frac{1}{x}dx\]
\[ \Rightarrow - 3 \times \frac{1}{v} = \log \left| x \right| + C\]
\[ \Rightarrow - \frac{3}{v} = \log \left| x \right| + C\]
\[\text{ Putting }v = \frac{y}{x},\text{ we get }\]
\[ \Rightarrow \frac{- 3x}{y} = \log \left| x \right| + C\]
\[\text{ Hence, }\frac{- 3x}{y} = \log \left| x \right| + C\text{ is the required solution }.\]