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y-x-dy-dx-a-y-2-dx-dy-




Question Number 129222 by bramlexs22 last updated on 14/Jan/21
 y−x (dy/dx) = a(y^2 +(dx/dy) )
$$\:\mathrm{y}−\mathrm{x}\:\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\mathrm{a}\left(\mathrm{y}^{\mathrm{2}} +\frac{\mathrm{dx}}{\mathrm{dy}}\:\right) \\ $$
Answered by bobhans last updated on 14/Jan/21
 y−xy′ = ay^2 +(a/(y′))   yy′−x(y′)^2 =ay^2 y′+a  (y−ay^2 )y′−x(y′)^2 =a   x(y′)^2 +(ay^2 −y)y′+a=0   y′=((y−ay^2 ± (√((ay^2 −y)^2 −4ax)))/(2x))
$$\:{y}−{xy}'\:=\:{ay}^{\mathrm{2}} +\frac{{a}}{{y}'} \\ $$$$\:{yy}'−{x}\left({y}'\right)^{\mathrm{2}} ={ay}^{\mathrm{2}} {y}'+{a} \\ $$$$\left({y}−{ay}^{\mathrm{2}} \right){y}'−{x}\left({y}'\right)^{\mathrm{2}} ={a} \\ $$$$\:{x}\left({y}'\right)^{\mathrm{2}} +\left({ay}^{\mathrm{2}} −{y}\right){y}'+{a}=\mathrm{0} \\ $$$$\:{y}'=\frac{{y}−{ay}^{\mathrm{2}} \pm\:\sqrt{\left({ay}^{\mathrm{2}} −{y}\right)^{\mathrm{2}} −\mathrm{4}{ax}}}{\mathrm{2}{x}} \\ $$$$ \\ $$

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