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Show that the given differential equation is homogeneous and solve each of them. ( x to the power 2 minus y to the power 2) dx plus 2xy dy equals 0

Show that the given differential equation is homogeneous
and solve each of them.

    Q4.    (x^2 - y^2)dx + 2xydy = 0

Answers (1)
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M manish

we can write it as;

\frac{dy}{dx}= -\frac{(x^{2}-y^{2})}{2xy} = F(x,y)\ (let\ say)...................................(i)

F(\lambda x,\lambda y) = \frac{(\lambda x)^{2}-(\lambda y)^{2}}{2(\lambda x)(\lambda y)} = \lambda ^{0}.F(x,y)
Hence it is a homogeneous equation

Now, to solve substitute y = vx
Differentiating on both sides wrt x
\frac{dy}{dx}= v +x\frac{dv}{dx}
                                 
Substitute this value in equation (i)

v+x\frac{dv}{dx} = \frac{ x^{2}-(vx)^{2}}{2x(vx)} =\frac{v^{2}-1}{2v}
\\x\frac{dv}{dx} =\frac{v^{2}+1}{2v}\\ \frac{2v}{1+v^{2}}dv=\frac{dx}{x}
integrating on both sides, we get 

\log (1+v^{2})= -\log x +\log C = \log C/x
                         \\= 1+v^{2} = C/x\\ = x^2+y^{2}=Cx.............[v =y/x]
This is the required solution.

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