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

Nov 18, 2016

$y = \left({C}_{2} + \arcsin \left(x\right)\right) x$

#### Explanation:

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

$x y {'}_{p} - {y}_{p} = \frac{x}{\sqrt{1 + {x}^{2}}}$ (2) so

$y = {y}_{h} + {y}_{p}$ (3)

The homogeneous solution is ${y}_{h} = {C}_{1} x$ The particular is obtained using the constant variation method due to Lagrange. So we make

${y}_{p} = C \left(x\right) x$ and substituting into (2) we obtain

$C ' \left(x\right) = \frac{1}{\sqrt{1 + {x}^{2}}}$

integrating $C \left(x\right)$ we get

$C \left(x\right) = \arcsin \left(x\right)$ and finally

$y = \left({C}_{2} + \arcsin \left(x\right)\right) x$