I am in need of directions to obtain the result (shown below) for the following problem:
For some constants, $\alpha$ and $Q$, I need to solve the following set of ODEs to find $b(x)$ and $u(x)$:
$$\frac{d}{dx} (b^2(x)u(x)) = 2\alpha bu$$
$$\frac{d}{dx} (b^2(x)u^2(x)) = \frac{Q}{u}$$
The result is presented as $b = \frac{6 \alpha}{5}x$ and $u = \frac{5}{6 \alpha} (\frac{9}{10} \alpha Q)^\frac{1}{3}x^{-\frac{1}{3}}$ in the paper by Morton et al.
If you set $(\mathbb{U},\mathbb{B})=(b^2u,b^2u^2)$ you can rewrite your system as $$\frac{d\mathbb{U}}{dx}=2\alpha\sqrt{\mathbb{B}}$$ $$\frac{d\mathbb{B}}{dx}=\frac{Q\mathbb{U}}{\mathbb{B}}$$ Dividing these equations yield $$\frac{d\mathbb{B}}{d\mathbb{U}}=\frac{Q\mathbb{U}}{2\alpha \mathbb{B}\sqrt{\mathbb{B}}}$$ which is separable. Can you finish?