Question

Solve the differential equation $x\frac{dy}{dx}+y+x+xy$=0.

Answer 1

Given,

$x\frac{dy}{dx}+y+x+xy=0$

$\frac{dy}{dx}+\frac{y}{x}+1+y=0$

$\frac{dy}{dx}+(\frac{1}{x}+1)y=-1$

$\frac{dy}{dx}+Py=Q$

$P=(\frac{1}{x}+1)\to I.F=e^{\int Pdx}=e^{\int (\frac{1}{x}+1)dx}=x{e}^x$

$Q=-1$

$y\times I.F=\int Q\times I.Fdx$

$y\times x{e}^x=\int -1\left(x{e}^x\right)dx$

$xy{e}^x=-e^{x}x+e^x+c$

$xy=1-x+\frac{c}{e^x}$