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

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

2021-02-25
Divide both sides by x and further simplify it
$$x/x dy/dx= y/x+(xe^(y/x))/x$$
$$dy/dx= y/x+e^(y/x)$$
Substitute $$y=xv => y/x=v$$
Differentiate with respect to x
$$dy/dx= v+x (dv)/dx$$
Hence, $$v+x (dv)/dx= v+e^v$$
Substract v from both sides and further simplify it
$$v+x (dv)/dx-v= v+e^v-v$$
$$x (dv)/dx= e^v$$
$$(dv)/e^v= dx/x$$
$$e^-v dv= dx/x$$
Integrate both sides
$$int e^v dv= int dx/x+c$$
$$-e^v= log x+log c$$
$$-e^v= log(xc)$$
$$e^v= -log(xc)$$
Taking log both sides
$$-v= log(-log(xc))$$
$$v= -log(-log(xc))$$ Substitute $$v= y/x$$
$$y/x= -log(-log(xc))$$
$$y= -x log(-log(xc))$$

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