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Question Number 169346 by Mastermind last updated on 29/Apr/22

Show that substituting y=vx,  x+y(dy/dx)=x(dy/dx)−y to a separable  equation for v and x and its   solution is log_e (x^2 +y^2 )=2tan^(−1) ((y/x))  +C    Mastermind

$${Show}\:{that}\:{substituting}\:{y}={vx}, \\ $$$${x}+{y}\frac{{dy}}{{dx}}={x}\frac{{dy}}{{dx}}−{y}\:{to}\:{a}\:{separable} \\ $$$${equation}\:{for}\:{v}\:{and}\:{x}\:{and}\:{its}\: \\ $$$${solution}\:{is}\:{log}_{{e}} \left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)=\mathrm{2}{tan}^{−\mathrm{1}} \left(\frac{{y}}{{x}}\right) \\ $$$$+{C} \\ $$$$ \\ $$$${Mastermind} \\ $$$$\: \\ $$

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