Find the particular solution of the differential equation `(dy)/(dx)=(x y)/(x^2+y^2)` given that `y=1\ w h e n\ x=0.`

Explanation

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Given: `frac{d y}{d x}=frac{x y}{x^{2}+y^{2}} ; y=1 { and } x=0`

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`{Let} f(x)=frac{x y}{x^{2}+y^{2}} Rightarrow f(lambda x, lambda y)=frac{(lambda x)(lambda y)}{(lambda x)^{2}+(lambda y)^{2}}=frac{lambda^{2} x y}{lambda^{2} x^{2}+lambda^{2} y^{2}}`

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`Rightarrow f(lambda x, lambda y)=frac{lambda^{2} x y}{lambda^{2}(x^{2}+y^{2}right)}=frac{x y}{x^{2}+y^{2}}=f(x, y)`

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`Rightarrow` the given differential equation is homogeneous.

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