1.

Find the general solution of the differential equation `(1+x^(2))(dy)/(dx)-x=2tan^(-1)x.`

Answer» The given differential equation may be written as
`(dy)/(dx)=(2tan^(-1)x)/((1+x^(2)))+(x)/((1+x^(2)))`
`rArr dy={(2tan^(-1)x)/((1+x^(2)))+(x)/((1+x^(2)))}dx " " ` [separating the variables]
`rArr int dy=int (2tan^(-1)x)/((1+x^(2)))dx+int (x)/((1+x^(2)))dx + C, " " ` where C is an arbitrary constant
`rArr y= 2int t dt + (1)/(2) int (2x)/((1+x^(2))) dx +C `
[putting `tan^(-1)x=t and (1)/((1+x^(2))) dx = dt ` in 1st integral]
`rArr y=t^(2)+(1)/(2)log|1+x^(2)| + C`
`rArr y=(tan^(-1)x)^(2)+(1)/(2)log|1+x^(2)|+C.`
Hence, `y=(tan^(-1)x)^(2)+(1)/(2)log|1+x^(2)|+C` is the required solution.


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