Number of differential equations and unknown functions in spherically symmetric black hole solution In General Relativity, when we are obtaining the Schwarzchild solution, we get from Einstein's equation three differential equations but only two unknown functions [A(r) and B(r)]:
$R_{00}=-\frac{A''}{2B}+\frac{A'}{4B}\left(\frac{A'}{A}+\frac{B'}{B}\right)-\frac{A'}{rB}=0,\\
R_{11}=\frac{A''}{2A}-\frac{A'}{4A}\left(\frac{A'}{A}+\frac{B'}{B}\right)-\frac{B'}{rB}=0,\\
R_{22}=\frac{1}{B}-1+\frac{r}{2B}\left(\frac{A'}{A}-\frac{B'}{B}\right)=0.$
Shouldn't the number of differential equations be equal to the number of unknown functions?
Here A(r) and B(r) are defined as
$ds^2=A(r)dt^2-B(r)dr^2-r^2(d\theta^2+\sin\theta^2d\phi^2).$
 A: You're correct that there are only two functions, meaning we only need two differential equations. Therefore one must be redundant, which is the case. Showering this can be slightly awkward though.
One trick is to take the derivative of the equations and work with these too. With the Einstein tensor it's a lot easier to do, but of course it can also be done with your differential equations here. I'm not sure there's any foolproof method of approaching this though.
A hand-wavy method is to take the derivative of all three equations (which I'll label $R'_{00}$, $R'_{11}$ and $R'_{22}$). Then if you solve $R'_{11}$ for $A'''(r)$, solve $R'_{22}$ for $B''(r)$ and solve $R_{11}$ for $A''(r)$, you can plug these into $R'_{00}$ and see it vanishes. Perhaps there's a better method here but not one that's obvious to me.
Alternatively, note that you can use just two of the equations to solve for $A(r)$ and $B(r)$ completely, then the third equation is automatically satisfied. Therefore one is made redundant, answering your question. However, when you have field equations that you can't find closed form solutions for, the method above is also useful for verifying the number of independent equations.
