Solve the differential equation (x-3y)dx-xdy=0

Solve the differential equation (x-3y)dx-xdy=0

Question
Solve the differential equation \((x-3y)dx-xdy=0\)

Answers (1)

2021-03-03
\((x-3y)-xdy/dx=0\)
\(-dy/dx-((x-3y)/x)=0\)
\(dy/dx-x/x+3y/x=0\)
\(dy/dx+3/x y=1\)
\(dy/dx+Py=Q\)
where \(P=3/x\), Q=1
\(I.F.= e^(int Pdx)\)
\(= e^(int 3/x dx)\)
\(= e^(3ln abs(x))\)
\(= e^(ln(x)^3)\)
\(= x^3\)
\(y(I.F.)= int Q(I.F.)+C\)
\(y(x^3)= int 1(x^3)+C\)
\(= x^4/4+C\)
\(y= x/4+C/x^3\)
\(= x/4+C_1\)
\(C_1= C/x^3\)
0

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