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Integrate the differential equation y′ = (-x)/y

Integrate the differential equation

    \[ y' = -\frac{x}{y}. \]


We separate the variables and integrate,

    \begin{align*}  y' &= -\frac{x}{y} & \implies && yy' &= -x \\  && \implies && \int y \, dy &= -\int x \, dx \\[9pt]  && \implies && \frac{1}{2}y^2 &= -\frac{1}{2}x^2 + C \\[9pt]  && \implies && x^2 + y^2 &= C. \end{align*}

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