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Integrate the differential equation x2y′ + xy + 2y2 = 0

Integrate the following differential equation:

    \[ x^2 y' + xy + 2y^2 = 0. \]


First, we have

    \[ x^2 y' + xy + 2y^2 = 0 \quad \implies \quad y' = -\frac{y}{x} - 2 \left( \frac{y}{x} \right)^2. \]

Then, we make the substitution v = \frac{y}{x}, which gives us y' = v + v'x and so,

    \begin{align*}  && y' &= -\frac{y}{x} - 2 \left( \frac{y}{x} \right)^2 \\[9pt]  \implies && v + v'x &= -v - 2v^2 \\[9pt]  \implies && -2(v^2  + v) &= v'x\\[9pt]  \implies && -\frac{1}{2} \int \frac{1}{v(v+1)} \, dv &= \int \frac{1}{x} \, dx \\[9pt]  \implies && \log (v+1) - \log |v| &= \log (x^2) + C \\[9pt]  \implies && \log \left| \frac{ \frac{y}{x} + 1}{\frac{y}{x}} \right| &= \log (x^2) + C \\[9pt]  \implies && 1 + \frac{x}{y} &= Cx^2 \\[9pt]  \implies && x &= y (Cx^2 - 1). \end{align*}

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