As part of my solution to a problem, I come to a point where I need to find the solutions to $-2\partial_{T}B\left(T\right)+\frac{3}{4}B\left(T\right)\left(A\left(T\right)^{2}+B\left(T\right)^{2}\right)=0$
$2\partial_{T}A\left(T\right)+\frac{3}{4}A\left(T\right)\left(B\left(T\right)^{2}+A\left(T\right)^{2}\right)=0$
where $\partial_{T}(f)$ is the derivative with respect to $T$.
It is possible that I made a mistake in the steps leading to this because I am supposed to be able to get a not-so-ugly solution for $A(T)$ and $B(T)$. Is there one that exists and I don't see it? I've tried the following: