# Solve and classify equation dy/dx= - (e^x+y)/(k+x+ye^x), y(0)=1

Solve and classify equation $dy/dx=-\left({e}^{x}+y\right)/\left(k+x+y{e}^{x}\right)$, y(0)=1
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Alix Ortiz
Solving the numerator part and denominator part
According the first order linear differential equation
$-{e}^{-}x+y$
$y={y}^{\prime }-y=-{e}^{x}$
$y=-{e}^{x}x+{c}_{1}{e}^{x}$
Solving the denominator part through first order linear differential equation
$dy/dx=k+x+y{e}^{x}$
${y}^{\prime }=k+x+y{e}^{x}$
${y}^{\prime }-{e}^{x}y=k+x$
solving the parts
$y=\left(-{e}^{x}x+{c}_{1}{e}^{x}\right)/\left(\left(k+x\right)/-{e}^{x}\right)$
This equation is in homogeneous form