xydydx=x^2 y^2 dydx=x^2 y^2xy #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; y=vx , dyd ...

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Particular Solution of a Differential Equation

xydydx=x2-y2

Question

$x y \frac{d y}{d x} = x^{2} - y^{2}$

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Solution

$x y \frac{d y}{d x} = x^{2} - y^{2}$
$\frac{d y}{d x} = \frac{x^{2} - y^{2}}{x y}$ $y = v x$ , $\frac{d y}{d x} = v + x \frac{d v}{d x}$
$v + x \frac{d v}{d x} = \frac{v}{y} - [\frac{x^{2} - v^{2} x^{2}}{x^{2} v}]$
$x \frac{d v}{d x} = \frac{x^{2}}{x^{2}} (\frac{1 - v^{2}}{v}) - v$
$x \frac{d v}{d x} = \frac{1 - v^{2} - v^{2}}{v}$
$\frac{1}{(- y)} \int (- y) v \frac{d v}{1 - 2 v^{2}} = \int \frac{d x}{x}$ .

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