Question

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

First order differential equations
ANSWERED
asked 2021-03-02
Solve the differential equation \((x-3y)dx-xdy=0\)

Answers (1)

2021-03-03

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

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