Discussion
The general solution of the differential equation \(ydx - xdy = y^{2}(xdy + ydx)\) is
Answer: E
The general solution is \(\frac{x}{y} = xy + C\).
Recognise the two exact differentials: \(xdy + ydx = d(xy)\) and \(\frac{ydx - xdy}{y^{2}} = d\left(\frac{x}{y}\right)\). Divide the equation by \(y^{2}\):
\[ \begin{aligned} \frac{ydx - xdy}{y^{2}} &= xdy + ydx \\ d\left(\frac{x}{y}\right) &= d(xy) \end{aligned} \]
Integrating, \(\frac{x}{y} = xy + C\). \(\frac{y}{x}\) has the quotient upside down, since \(d\left(\frac{y}{x}\right) = \frac{xdy - ydx}{x^{2}}\), with the opposite sign and \(x^{2}\) underneath.
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