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

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

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
$$I.F.= e^(int Pdx)$$
$$= e^(int 3/x dx)$$
$$= e^(3ln abs(x))$$
$$= e^(ln(x)^3)$$
$$= x^3$$
$$y(I.F.)= int Q(I.F.)+C$$
$$y(x^3)= int 1(x^3)+C$$
$$= x^4/4+C$$
$$y= x/4+C/x^3$$
$$= x/4+C_1$$
$$C_1= C/x^3$$

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