1. **Stating the problem:** We need to find the function $b(r)$ satisfying the equation $$b(r)\frac{\alpha}{r^3} + E^2 = -w b(r)^u \left(-\alpha \frac{b'(r)}{r^3} - E^2\right)$$ where $$E^2 = k a r^2 \frac{5 + a r^2}{(1 + a r^2)^3}$$ 2. **Rewrite the equation:** Substitute $E^2$ into the equation: $$b(r)\frac{\alpha}{r^3} + k a r^2 \frac{5 + a r^2}{(1 + a r^2)^3} = -w b(r)^u \left(-\alpha \frac{b'(r)}{r^3} - k a r^2 \frac{5 + a r^2}{(1 + a r^2)^3}\right)$$ 3. **Simplify the right side:** Distribute the negative sign inside the parentheses: $$-w b(r)^u \left(-\alpha \frac{b'(r)}{r^3} - E^2\right) = -w b(r)^u \left(-\alpha \frac{b'(r)}{r^3}\right) - (-w b(r)^u E^2) = w b(r)^u \alpha \frac{b'(r)}{r^3} + w b(r)^u E^2$$ 4. **Rewrite the full equation:** $$b(r)\frac{\alpha}{r^3} + E^2 = w b(r)^u \alpha \frac{b'(r)}{r^3} + w b(r)^u E^2$$ 5. **Group terms:** Bring all terms to one side: $$b(r)\frac{\alpha}{r^3} + E^2 - w b(r)^u E^2 = w b(r)^u \alpha \frac{b'(r)}{r^3}$$ 6. **Isolate $b'(r)$:** $$w b(r)^u \alpha \frac{b'(r)}{r^3} = b(r)\frac{\alpha}{r^3} + E^2 (1 - w b(r)^u)$$ Divide both sides by $w b(r)^u \alpha / r^3$: $$b'(r) = \frac{r^3}{w b(r)^u \alpha} \left(b(r)\frac{\alpha}{r^3} + E^2 (1 - w b(r)^u)\right) = \frac{b(r)}{w b(r)^u} + \frac{r^3 E^2 (1 - w b(r)^u)}{w b(r)^u \alpha}$$ Simplify the first term: $$\frac{b(r)}{w b(r)^u} = \frac{b(r)^{1-u}}{w}$$ 7. **Final differential equation:** $$b'(r) = \frac{b(r)^{1-u}}{w} + \frac{r^3 E^2 (1 - w b(r)^u)}{w b(r)^u \alpha}$$ 8. **Substitute $E^2$ explicitly:** $$b'(r) = \frac{b(r)^{1-u}}{w} + \frac{r^3 k a r^2 \frac{5 + a r^2}{(1 + a r^2)^3} (1 - w b(r)^u)}{w b(r)^u \alpha} = \frac{b(r)^{1-u}}{w} + \frac{k a r^5 (5 + a r^2)}{(1 + a r^2)^3} \frac{1 - w b(r)^u}{w b(r)^u \alpha}$$ 9. **Interpretation:** This is a nonlinear first-order ODE for $b(r)$: $$b'(r) = \frac{b(r)^{1-u}}{w} + \frac{k a r^5 (5 + a r^2)}{(1 + a r^2)^3} \frac{1 - w b(r)^u}{w b(r)^u \alpha}$$ 10. **Solution approach:** This ODE generally requires numerical methods or special assumptions on parameters $u, w, \alpha, k, a$ to solve explicitly. **Summary:** We derived the differential equation for $b(r)$ from the given implicit relation and expression for $E^2$. The solution depends on parameters and may need numerical solving.