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Verify that the given functions (explic...

Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:`x+y=tan^(-1)y` : `y^2y^(prime)+y^2+1=0`

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Given, `x+y=tan^(-1)y`…………..`(1)`
`implies (d)/(dx)(x+y)=(d)/(dx)tan^(-1)y`
`implies 1+y'=(1)/(1+y^(2))(y')`
`implies (1+y')(1+y^(2))=y'`
`implies 1+y^(2)+y'+y^(2)y'=y'`
`implies 1+y^(2)+y^(2)y'=0`
Therefore, `x+y=tan^(-1)y` is the solution of given differential equation.
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