Question

Solve:

dy/dx =2xy / ( x2 - y2 )

The answer given in the book is :y = C ( x2 + y2 )

Answer 1

Please see the attachment

Answer 2

Answer:

$y=c(x^2+y^2)$

Step-by-step explanation:

We are given that

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

It is a homogeneous function equation

Substitute y=vx

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

Then we get

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

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

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

$x\frac{dv}{dx}=\frac{2v-v+v^3}{1-v^2}$

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

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

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

Integrating on both sides then we get

$ln v-ln(1+v^2)=lnx+lnc$

Substitute $v=\frac{y}{x}$

Then we get

$ln\frac{y}{x}-ln(1+\frac{y^2}{x^2})=lncx$

$ln\frac{y}{x}\cdot\frac{x^2}{x^2+y^2}=ln cx$

Cancel ln on both side

$\frac{yx}{x^2+y^2}=cx$

$\frac{y}{x^2+y^2}=c$

$y=c(x^2+y^2)$