Error in solution to differential equation? I'm trying to solve $w^5L\frac{dL}{dw} = 1 - w^4L^2$. I attempted the substitution $M = \frac12w^5L^2$, so that $M' = \frac52w^4L^2 + w^5L'L$. Then
$$
w^5L'L = M' - \frac52 w^4L^2 = M' - \frac{5M}w \\
1 - w^4L^2 = 1 - \frac{2M}w
$$
so the equation becomes
$$
M' = 1 + \frac{3M}w
$$
which can be solved to give
$$
M = c_1w^3 - \frac12w
$$
and so
$$
w^5L^2 = c_1w^2(2w - 1) \\
L = \pm\sqrt{\frac{c_1(2w-1)}{w^3}}
$$
however, apparently, I'm supposed to get
$$
L = \pm\frac{\sqrt{c_1w^2 - 1}}{w^2}
$$
so it seems I'm missing a factor of $w$ somewhere, or something, but I'm not quite sure. Where am I going wrong?

$\frac{1}{2}w^5L^2=-\frac{w}{2}+Cw^3$
Multiply 2 on both sides and then divide $m^5$ on both sides gives:
$L^2=\frac{-w+Kw^3}{w^5}=\frac{-1+kw^2}{w^4}$
