Solve differential equation xydy−y^2dx= (x+y)^2 e^(−y/x)

Question
Solve differential equation \(xydy−y^2dx= (x+y)^2 e^(−y/x)\)

Answers (1)

2020-11-08
\(xydy-y^2dx= (x+y)^2 e^(−y/x)\)
\(xy dy/dx-y^2= (x+y)^2 e^(−y/x)\) (1)
Put \(y/x= t => dy/dx= y+x (dt)/dx\)
Now the equation (1) becomes
\(x*xt [t+x (dt)/dx]-x^2t^2= x^2(1+t)^2 e^-t\)
\(t[t+x (dt)/dx]-t^2= (1+t)^2 e^t\)
\(xt (dt)/dx= (1+t)^2 e^t\)
\((e^t tdt)/(1+t)^2= dx/x\)
\((e^t((t+1)-1)dt)/(1+t)^2= dx/x\)
\(e^t (1/(1+t)-1/(1+t)^2)dt= dx/x\) (2)
Integrating on the both sides
\(int e^t(1/(1+t)-1/(1+t)^2)dt= int dx/x\)
Use the standard formula of integration
\(int e^x(f(x)+f'(x))dx= e^x f(x)+C\)
\(e^t/((1+t))= ln(x)+log(C)\)
\(e^(y/x)/(1+y/x)= ln(Cx)\)
\(e^(y/x)= (1+y/x)ln(Cx)\)
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