Solve the differential equation #x y'-y=x/sqrt(1+x^2)# ?

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Answer 1 · 2016-11-18T19:13:15

#y = (C_2+arcsin(x))x#

This is a linear nonhomogeneous differential equation. The solution is obtained as the sum of the homogeneous solution

#x y'_h-y_h=0# (1)

and the particular solution

#x y'_p-y_p=x/sqrt(1+x^2)# (2) so

#y = y_h + y_p# (3)

The homogeneous solution is #y_h=C_1x# The particular is obtained using the constant variation method due to Lagrange. So we make

#y_p=C(x)x# and substituting into (2) we obtain

#C'(x)=1/sqrt(1+x^2)#

integrating #C(x)# we get

#C(x)=arcsin(x)# and finally

#y = (C_2+arcsin(x))x#