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Find an implicit formula satisfied by solutions of xyy′ = 1 + x2 + y2 + x2y2

Assume solutions of the equation

    \[ xyy' = 1 + x^2 + y^2 + x^2 y^2 \]

exist and find an implicit formula satisfied by these solutions.


This is a first-order separable equation. We compute

    \begin{align*}  xyy' = 1 + x^2 + y^2 + x^2 y^2 && \implies && y' \left( \frac{y}{1+y^2} \right) &= x + \frac{1}{x} \\  && \implies && \int \frac{y}{1+y^2} \, dy &= \int \left( x + \frac{1}{x} \right) \, dx \\  && \implies && \frac{1}{2} \log (1+y^2) &= \frac{1}{2} x^2 + \log |x| + C \\  && \implies && 1+y^2 &= Cx^2 e^{x^2}.  \end{align*}

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