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Given this \(F = p(x)y^{'2}-q(x)y^2+2f(x)y\). What would be the integral of \(f(x)\) and \(q(x)\)?

\begin{align*}

f(x) - q(x)y - \frac{d}{dx}\left[p(x)y'\right] &= 0\\

\frac{d}{dx}\left[p(x)y'\right] &= f(x) - q(x)y\\

y'p(x) &= \int f(x)dx - y\int q(x)dx

\end{align*}

\begin{align*}

f(x) - q(x)y - \frac{d}{dx}\left[p(x)y'\right] &= 0\\

\frac{d}{dx}\left[p(x)y'\right] &= f(x) - q(x)y\\

y'p(x) &= \int f(x)dx - y\int q(x)dx

\end{align*}

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