<s>[INST] Solving the differential equation: $f(x)yy'=(y')^2-0.5$ I am trying to solve this equation:
$f(x)yy'=(y')^2-0.5$
I have already tried traditional methods...
Any ideas?
 [/INST] We have
\begin{align*}
&y y^\prime = (y^\prime)^2 - 0.5 \\
& (y^\prime)^2 - y y^\prime = 0.5 \\
& (y^\prime)^2 - y y^\prime + \frac{1}{4}y^2 = 0.5 + \frac{1}{4} y^2 \\
& \left(y^\prime - \frac{1}{2} y \right)^2 = 0.5 + \frac{1}{4} y^2 \\
& y^\prime  = \frac{1}{2} \left(y  \pm \sqrt{1 + y^2} \right)
\end{align*}
You can solve this implicity using separation of variables.
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