Question

Solve the following initial value problem:
(xy − y²) dx − x² dy = 0, y(1) = 1

Answer 1

(xy − y²) dx − x² dy = 0, y(1) = 1
This is an homogenous equation, put y = vx
\[\frac{dy}{dx} = v + x\frac{dv}{dx}\]
\[\left( xy - y^2 \right) = x^2 \left( \frac{dy}{dx} \right)\]
\[\left( v x^2 - v^2 x^2 \right) = x^2 \left( v + x\frac{dv}{dx} \right)\]
\[v x^2 \left( 1 - v \right) = x^2 \left( v + x\frac{dv}{dx} \right)\]
\[v\left( 1 - v \right) = v + x\frac{dv}{dx}\]
\[v - v^2 = v + x\frac{dv}{dx}\]
\[ - v^2 = x\frac{dv}{dx}\]
\[ - \frac{1}{x}dx = \frac{1}{v^2}dv\]
On integrating both sides we get,
\[- \int\frac{1}{x}dx = \int\frac{1}{v^2}dv\]
\[ - \log_e x = \frac{v^{- 2 + 1}}{- 2 + 1} + c\]
\[ - \log_e x = \frac{v^{- 1}}{- 1} + c\]
\[ - \log_e x = - \frac{1}{v} + c\]
\[ - \log_e x = - \frac{1}{v} + c\]
\[\frac{x}{y} - \log_e x = c\]
\[\text{ As }y\left( 1 \right) = 1\]
\[\frac{1}{1} - \log_e 1 = c\]
\[ \Rightarrow c = 1\]