Question

solve the differential (1+x^2) dy/dx +y = e^tan-1 x​

Answer 1

Answer:

The given differential equation may be written as

dydx+1(1+x2)⋅y=etan−1x(1+x2)

This is of the formm dydx+Py=Q , where P=1(1+x2)andQ=etan−1x(1+x2)

Thus, the given equation is linear.

IF=e∫Pdx=e∫1(1+x2)dx=etan−1x.

So, the required solution is given by

y×IF=∫∣(Q×(IF)∣dx+C,

i.e, y×etan−1x=∫{etan−1x(1+x2)×etan−1x}dx+C

=∫e2tan−1x(1+x2)dx+C

=∫e2tdt+C, where tan−1x=t

=12e2t+C=12e2tan−1x+C.

Hence, y=12etan−1x+Ce−tan−1x is the required solution.