First slide

Methods of solving first order first degree differential equations

Question

$The general solution of the differential equation y (1 - xy) dx = x (1 + xy) dy is$

Moderate

A

$ln (\frac{x}{y}) =xy+c$
B

$ln (xy) =xy+c$
C

$ln (y/x) =xy+c$
D

$log (xy) =x+y+c$

Solution

$y d x - x y^{2} d x = x d y + x^{2} y d y$
$\begin{array}{l} ydx - xdy = xy (xdy + ydx) \\ \frac{y d x}{x y} - \frac{x d y}{x y} = x d y + y d x \\ \Rightarrow \int (\frac{dx}{x} - \frac{dy}{y}) = \int d (xy) \\ \Rightarrow \log \frac{x}{y} = xy + c \end{array}$

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