The solution of the differential equation `(1+y^2)+(x-e^(tan^-1)y)` (dy)/(dx)=0,` is (1) `(x-2)=k e^(tan^-1y)` (2) `2xe^(2tan^-1y) =e^(2tan^-1y)+k` (3) `xe^(tan^-1y)=tan^-1y+k` (4) `xe^(2tan^-1y)=e^(tan^-1y)+k`

Explanation

`z=tan^(-1)y`

`dz/dy=1/(1+y^2),1+y^2=dy/dz`

`dy/dz+(x-e^z)dy/dx=0`

`(dy/dz)/(dy/dx)+(x-e^2)=0`

`dx/dz+x-e^2=0`

`dx/dz+x=e^z`

`e^2*dx/dz+e^2x=e^(2z)`

`d/dz(x*e^z)=e^(2z)`

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