Doubt from Differential Equation

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homogenous differential equation @Ishan_2020
put y=vx

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Done that final integration me dikatt aa rahi hai

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Multiply by 2xy to get

y^2(y^2-2x^2) 2x \ \mathrm{dx}+x^2 (2y^2-x^2) 2y\ \mathrm{dy}=0

So let x^2=u,y^2=v so that v(v-2u) \ \mathrm{du}+u(2v-u) \ \mathrm{dv}=0

or (v^2 \mathrm{du} + 2uv \ \mathrm{dv)}-(u^2dv+2vu \ \mathrm{du})=0

or \mathrm{duv^2} = \ \mathrm{du^2v}

so that uv^2=u^2v+c \Rightarrow x^2y^4=x^4y+c or xy \sqrt{y^2-x^2}=\text{constant}

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Shortest and unique...