I'm trying to understand the setup given in Multiscale analytical solutions and homogenization of n-dimensional generalized elliptic equations, pg 41:
Given the boundary value problem:
$$\frac{d}{dx}\left( -K(x)\frac{du}{dx}\right)=0\text{ in } \Omega\equiv(0,1)$$ $$u(0)=0$$ $$u(1)=1$$ where $K(x)$ is a smooth, strictly positive function in $\Omega$.
We can consider the quantity $q^0 \equiv -K(x)\frac{du}{dx}$ to be a constant in $\Omega$, since the equation specifically states that $\frac{d}{dx}\left( q^0\right)=0$. I understand that we can write:
$$\frac{du}{dx}=-\frac{q^0}{K(x)}$$ $$\Rightarrow u=\int_0^x -\frac{q^0}{K(\tau)} d\tau$$.
But the author claims that we can further rewrite this as:
$$u=K^0\int_0^x\frac{d\tau}{K(\tau)}$$ where $$K^0=-q^0=\frac{1}{\int_0^1 \frac{d\tau}{K(\tau)}}$$.
It is not immediately apparent from the given information that we can deduce that $-q^0=\frac{1}{\int_0^1 \frac{d\tau}{K(\tau)}}$. How could I have derived this from the given information?
Plug the second boundary condition $u(1)=1$ into your formula $$u=\int_0^x -\frac{q^0}{K(\tau)} d\tau$$