1.

Find the equation of the curve that passes through the point (1, 2) and satisfies the differential equation `(dy)/(dx) =(-2xy)/((x^(2)+1)).`

Answer» We have
`(dy)/(dx) =(-2xy)/((x^(2)+1))`
`rArr (dy)/(y)=(-2x)/((x^(2)+1))dx" " `[on separating the variables]
`rArr int (dy)/(y)=int(-2x)/((x^(2)+1))dx" " `[integrating both sides]
`rArr log y = -log(x^(2)+1)+logC,` where log C is an arbitrary constant
`rArr log y + log (x^(2)+1)=log C`
`rArr log {y(x^(2)+1)}=log C`
`rArr y(x^(2)+1)=C " " `...(i)
Now, it is given that the curve passes through (1, 2).
So, putting x = 1 and y = 2 in (i), we get C = 4
`therefore y=(x^(2) +1)=4` is the required equation of the curve.


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