# Solve the differential equation x dy/dx= y+xe^(y/x), y=vx

Solve the differential equation $xdy/dx=y+x{e}^{\left(}y/x\right)$, y=vx
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krolaniaN

Divide both sides by x and further simplify it
$x/xdy/dx=y/x+\left(x{e}^{y/x}\right)/x$
$dy/dx=y/x+{e}^{\left(}y/x\right)$
Substitute $y=xv=>y/x=v$
Differentiate with respect to x
$dy/dx=v+x\left(dv\right)/dx$
Hence, $v+x\left(dv\right)/dx=v+{e}^{v}$
Substract v from both sides and further simplify it
$v+x\left(dv\right)/dx-v=v+{e}^{v}-v$
$x\left(dv\right)/dx={e}^{v}$
$\left(dv\right)/{e}^{v}=dx/x$
${e}^{-}vdv=dx/x$
Integrate both sides
$\int {e}^{v}dv=\int dx/x+c$
$-{e}^{v}=logx+logc$
$-{e}^{v}=log\left(xc\right)$
${e}^{v}=-log\left(xc\right)$
Taking log both sides
$-v=log\left(-log\left(xc\right)\right)$
$v=-log\left(-log\left(xc\right)\right)$ Substitute $v=y/x$
$y/x=-log\left(-log\left(xc\right)\right)$
$y=-xlog\left(-log\left(xc\right)\right)$