A solution curve of the differential equation $$x2+xy+4x+2y+4dydx-y2=0$$,x$$0$$, passes through the point $$1$$,$$3$$. Then the solution curve is

Question

A solution curve of the differential equation $$x2+xy+4x+2y+4dydx-y2=0$$,x>$$0$$, passes through the point $$1$$,$$3$$. Then the solution curve is

Infinity Learn · Accepted Answer

Given equation is $$x+22+yx+2dydx-y2=0$$  Let $$x+2=X$$, $$y=YX2+XYdY$$−$$Y2dX=0dYdX=Y2X2+XYY=vX$$⇒$$dvdX+v=v2v+1$$             ⇒$$XdvdX=$$−$$vv+1$$          ⇒ $$v+1vdv=$$−$$1XdX$$         ⇒$$v+logv=$$−$$logX+logc$$ ⇒$$YX+logY=logc$$⇒ $$yx+2+logy=logcit$$ passes through $$1$$,$$3$$⇒$$1+log3=logctherefore$$ $$yx+2+logy=1+log3option$$ $$1$$: if $$y=x+2$$ then $$x+2x+2+logy=1+log3$$    $$logy=log3$$  ⇒$$y=3$$ hence $$1$$,$$3$$ is common $$solutionoption3$$ : if $$y=x+22$$ then $$x+22x+2+logx+22=1+log3$$⇒$$x+2+logx+22=1+log3$$, no value of x satisfies.option $$4$$: if $$y=x+32$$ then $$x+32x+2+logx+32=1+log3$$,no value of x satisfies.

$A solution curve of the differential equation (x^{2} + x y + 4 x + 2 y + 4) \frac{d y}{d x} - y^{2} = 0, x > 0, passes$ through the point $(1, 3)$ . Then the solution curve is

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A solution curve of the differential equation x2+xy+4x+2y+4dydx-y2=0,x>0, passes through the point 1,3. Then the solution curve is

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$A solution curve of the differential equation (x^{2} + x y + 4 x + 2 y + 4) \frac{d y}{d x} - y^{2} = 0, x > 0, passes$ through the point $(1, 3)$ . Then the solution curve is