Necessary conditions for transforming a system of O.D.Es to a single O.D.E. Considering the system:
$$u' = a(x)u + b(x)v$$
$$v' = c(x)u + d(x)v$$
I transform it to the second order O.D.E.(please check me on this because I might have mistakes): 
$$\displaystyle u'' - \left (\frac{b'(x)}{b(x)} + a(x) + b(x) \right )u' + \left ( \frac{a(x)b'(x)}{b(x)} - a'(x) - c(x)b(x) + a(x)d(x) \right )u = 0$$
In the process, I am allowed to assuming as much differentiability as I want (on every function). Are there any other necessary conditions on the coefficients $a, b, c, d$ to do the above transformation?

$v=(u'-au)/b$, 
$v'=(-b'(u'-au)+b(u''-a'u-au'))/b^2=cu+d(u'-au)/b$, 
$-b'u'+ab'u+bu''-a'bu-abu'=b^2cu+bdu'-abdu$, 
$bu''-(bd+b'+ab)u'+(-b^2c+abd+ab'-a'b)u=0$ 
which is almost what you have --- I have a $du'$ where you have $bu'$. 
The way you have written it, you need to worry about the possibility that $b(x)=0$ for some $x$ in the domain. 
