Let $$C1$$ be the curve obtained by the solution of the differential equation $$2xydydx=y2$$−$$x2Let$$ the curve $$C2$$ be the solution of $$2xyx2$$−$$y2=dydx$$.If both curves pass through the point $$1$$,$$1$$,then the area enclosed by the curve $$C1$$ and $$C2$$ is equal to

Question

Let $$C1$$  be the curve obtained by the solution of the differential equation $$2xydydx=y2$$−$$x2Let$$ the curve $$C2$$ be the solution of $$2xyx2$$−$$y2=dydx$$.If both curves pass through the point $$1$$,$$1$$,then the area enclosed by the curve $$C1$$  and $$C2$$  is equal to

Infinity Learn · Accepted Answer

The given differential equation $$dydx=y2-x22xy$$, it is homogeneous equation.  Put $$y=vxv+xdvdx=v2x2-x22vx2=v2-12vxdvdx=v2-1-2v22v=-v2+12v$$⇒$$2vv2+1dv=-dxxlnv2+1=-$$ℓ$$nx+lnc$$⇒$$v2+1=cx$$⇒$$y2x2+1=cx$$⇒$$x2+y2=cx$$ If pass through (1$$,$$1)  ∴$$x2+y2-2x=0Similarly$$ second differential equation is $$dxdy=x2$$−$$y22xyEquation$$ of curve is $$x2+y2$$−$$2y=0$$ Area of the region required is equal to the double the $$($$ area of the quarter circle $$-$$ area of the triangle $$)=14$$×π×$$12$$−$$12$$×$$1$$×$$1$$×$$2=$$π$$2$$−$$1$$ square units

Let C1 be the curve obtained by the solution of the differential equation 2xydydx=y2−x2Let the curve C2 be the solution of 2xyx2−y2=dydx.If both curves pass through the point 1,1,then the area enclosed by the curve C1 and C2 is equal to

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