The solution of x²+y²-2xydy/dx=0 is :

Correct Answer is : x2−y2=cxx²-y²=cxx2−y2=cx

Correct Answer is : x2−y2=cxx²-y²=cxx2−y2=cx

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BITSAT 2005 Solved Question Paper

Section: Mathematics

Question : 28 of 150

Marks: +1, -0

The solution of

x^{2}+y^{2}-2xy\frac{dy}{dx}=0

Solution:

Given equation is

x^{2}+y^{2}-2xy\frac{dy}{dx}=0

⇒

\frac{dy}{dx}=\frac{x^{2}+y^{2}}{2xy}

.......(i)

This is a homogeneous equation

\therefore

we put

y=vx

and

\frac{dy}{dx}=v+x\frac{dv}{dx}

The Eq. (i) is reduces to

v+x\frac{dv}{dx}=\frac{x^{2}(1+v^{2})}{2x^{2}v}

⇒

x\frac{dv}{dx}=\frac{1+v^{2}}{2v}-v=\frac{1-v^{2}}{2v}

⇒

-\frac{2v}{1-v^{2}}dv=-\frac{dx}{x}

On integrating both sides, we get

\log(1-v^{2})=-\log x+\log c

\Rightarrow \log(x^{2}-y^{2})-2\log x=-\log x+\log c

\Rightarrow \log(x^{2}-y^{2})=\log x c

\Rightarrow x^{2}-y^{2}=xc

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