How would you solve this differential equation: $\frac{d^2y}{dx^2} = \frac{100}{y}$? The equation is:

$$\frac{d^2y}{dx^2} = \frac{100}{y}$$

Also if $y = f(x)$, 

$f'(0) = 0$ and $f(0) = 10$

How would you solve this for $y$?
$$y''=\frac {100} y$$
is autonomous (ie $x$ is not present in the equation). The standard procedure therefore is to substitute $u(y)=y'$ with
$$ y'' = \frac{du}{dx} = \frac{du}{dy}\frac{dy}{dx} = u'u $$
solve
$$ u'u = 100/y $$
with $u'u = \frac{1}{2}(u^2)'$ we get
$$ u(y)^2 = 2\left(c_1 + 100\ln(y)\right). $$
This equation "can be integrated"
$$ y' = \pm\sqrt{2\left(c_1 + 100\ln(y)\right)} $$
$$ \Rightarrow \int\frac{1}{\sqrt{2\left(c_1 + 100\ln(y)\right)}}dy = \mp\int 1\,dx $$
This is the point where it gets ugly...

The integral can be evaluated to
$$ \sqrt{\frac{\pi}{200}} e^{-c_1/100}\operatorname{erfi}\left(\sqrt{\frac{c_1}{100}+\ln(y)}\right) + c_2 = \mp x $$
where $\operatorname{erfi}$ is the imaginary error function. Solve for $y$ and you are done...
Wolfram alpha gives this result which shows it is not easy to find.