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Mathematics Question on General and Particular Solutions of a Differential Equation

If 1−x6+1−y6=a(x3−y3),\sqrt{1-x^{6} }+ \sqrt{1-y^{6}} =a\left(x^{3} -y^{3}\right) , then y2dydx=y^{2} \frac{dy}{dx} =

A

1−y61−x6\sqrt{\frac{ 1- y^6}{ 1 - x^6}}

B

x1−y61−x6 x \sqrt{\frac{ 1- y^6}{ 1 - x^6}}

C

x21−y61−x6x^2 \sqrt{\frac{ 1- y^6}{ 1 - x^6}}

D

1x21−y61−x6\frac{1}{x^2} \sqrt{\frac{ 1- y^6}{ 1 - x^6}}

Answer

x21−y61−x6x^2 \sqrt{\frac{ 1- y^6}{ 1 - x^6}}

Explanation

Solution

Given equation is
1−x6+1−y6=a(x3−y3)\sqrt{1-x^{6}}+\sqrt{1-y^{6}}=a\left(x^{3}-y^{3}\right)
⇒1−x6+1−y6x3−y3=a\Rightarrow \frac{\sqrt{1-x^{6}}+\sqrt{1-y^{6}}}{x^{3}-y^{3}}=a
On differentiating both sides w.r.t., xx, we get
[(x3−y3)(−6x521−x6−6y521−y6dydx)−(1−x6+1−y6)(3x2−3y2dydx)](x3−y3)2=0\frac{\left[\left(x^{3}-y^{3}\right) \left(\frac{-6x^{5}}{2\sqrt{1 -x^{6}}} -\frac{6y^{5}}{2\sqrt{1 -y^{6}}} \frac{dy}{dx}\right) - \left(\sqrt{1 -x^{6}}+\sqrt{1- y^{6}}\right)\left(3x^{2} -3y^{2} \frac{dy}{dx}\right)\right]}{\left(x^{3}-y^{3}\right)^{2}}=0
⇒(y2(1−x6+1−y6)−y5(x3−y3)1−y6)dydx\Rightarrow\left(y^{2}\left(\sqrt{1-x^{6}}+\sqrt{1-y^{6}}\right)-\frac{y^{5}\left(x^{3}-y^{3}\right)}{\sqrt{1-y^{6}}}\right) \frac{d y}{d x}
=x2(1−x6+1−y6)+x51−x6(x3−y3)=x^{2}\left(\sqrt{1-x^{6}}+\sqrt{1-y^{6}}\right)+\frac{x^{5}}{\sqrt{1-x^{6}}}\left(x^{3}-y^{3}\right)
⇒y2dydx[1−x6+1−y6+(1−y6)−y3x3+y61−y6]\Rightarrow y^{2} \frac{d y}{d x}\left[\frac{\sqrt{1-x^{6}}+\sqrt{1-y^{6}}+\left(1-y^{6}\right)-y^{3} x^{3}+y^{6}}{\sqrt{1-y^{6}}}\right]
=x2[(1−x6)+1−y61−x6+x6−x3y31−x6]=x^{2}\left[\frac{\left(1-x^{6}\right)+\sqrt{1-y^{6}} \sqrt{1-x^{6}}+x^{6}-x^{3} y^{3}}{\sqrt{1-x^{6}}}\right]
⇒y2dydx[1−x61−y6+1−x3y31−y6]\Rightarrow y^{2} \frac{d y}{d x}\left[\frac{\sqrt{1-x^{6}} \sqrt{1-y^{6}}+1-x^{3} y^{3}}{\sqrt{1-y^{6}}}\right]
=x2[1+1−y61−x6−x3y31−x6]=x^{2}\left[\frac{1+\sqrt{1-y^{6}} \sqrt{1-x^{6}}-x^{3} y^{3}}{\sqrt{1-x^{6}}}\right]
⇒y2dydx=x21−y61−x6\Rightarrow y^{2} \frac{d y}{d x}=x^{2} \sqrt{\frac{1-y^{6}}{1-x^{6}}}