Thus, the velocity profile was:
[ r \frac{dv}{dr} + v = 3r^3 ]
[ \frac{d}{dr}(r v) = 3r^3 ]
[ P = \int_{0}^{R} v(r) , dr = \int_{0}^{4} \frac{3}{4} r^3 , dr ] Integral calculus including differential equations
Integrating both sides with respect to ( r ): Thus, the velocity profile was: [ r \frac{dv}{dr}
The integrating factor ( \mu(r) ) was:
Lyra raced to the control platform. She encoded the function into the harmonic resonators, and as the monsoon winds arrived, the great whirlpool shuddered—then dissolved into a spiral of calm, glimmering water. dr = \int_{0}^{4} \frac{3}{4} r^3