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

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

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