I am doing physics research and have come across the following boundary value problem.
I have functions $f(x), g(x), t(x), u(x)$ which follow:
\begin{align} \frac{\text{d} f}{\text{d} x} &= x^{2} t\\ \frac{\text{d} f}{\text{d} x} &= \frac{x^{2} g}{u}\\ \frac{\text{d} g}{\text{d} x} &= -\frac{f t}{x^{2}}\\ \frac{\text{d} \ln g}{\text{d} x} &=-\frac{f}{u x^{2}}\\ \end{align} subject to the boundary conditions \begin{align} f=0 \quad &\text{at} \quad x=0\\ g=c \quad &\text{at} \quad x=1 \end{align} where $c$ is a constant.
Given $u$, I can solve for $f,g$.
If I enforce that $f=1$ at $x=1$, can this be used to obtain a constraint on $u$?
Any advice or directions to look in would be greatly appreciated.
$f(0) = 0, f(1) = 1$ puts a restraint on $t$ that $$\int_0^1 x^2t(x)\, dx = 1$$
For any such $t$, we then get $$f(x) = \int_0^x s^2t(s)\,ds$$ and $$g(x) = c - \int_1^x s^{-2}f(s)t(s)\,ds$$ and finally $$\begin{align}u(x) &= \frac {g(x)}{t(x)}\\ &= \frac 1{t(x)}\left(c - \int_1^x s^{-2}t(s)\int_0^s r^2t(r)\,dr\,ds\right)\end{align}$$
How to directly express the restraint that $\int_0^1x^2t(x)\, dx = 1$ puts on $u$ is not clear.