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

Answers (1)

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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