Solve ` (1 + x ^(2)) (dy )/(dx) + y = e ^( tan ^(-1) x )`.

Explanation

The given differential equation may be written as
` (dy)/(dx) + (1)/(( 1 + x ^(2)) ) * y = ( e ^( tan ^(-1) x )) /( ( 1+ x ^(2))) `
This is of the formm ` (dy )/(dx) + P y = Q ` , where `P = (1)/( ( 1 + x ^(2))) and Q = (e ^(tan ^(-1) x ))/( (1 + x ^(2)))`
Thus, the given equation is linear.
`IF = e ^(int Pdx) = e ^( int (1)/(( 1 + x ^(2))) dx) = e ^(tan ^(-1) x )`.
So, the required solution is given by
`y xx IF = int |( Q xx ( IF) | dx + C`,
i.e, `y xx e ^( tan ^(-1) x ) = int { ( e ^(tan ^(-1)x))/( (1 + x ^(2))) xx e ^(tan ^(-1)x)} dx + C `
` " " = int ( e ^(2tan ^(-1) x ))/( (1 + x ^(2))) dx + C `
` " " = int e ^(2t)dt + C`, where ` tan ^(-1)x = t `
` = (1)/(2) e ^( 2t ) + C = (1)/(2) e ^(2tan^(-1)x) + C`.
Hence, `y = (1)/(2) e^(tan ^(-1) x ) + C e^(-tan ^(-1) x ) ` is the required solution.