Integral Calculus Including Differential Equations «EXTENDED →»

[ \frac{dv}{dr} + \frac{1}{r} v = 3r^2 ]

Lyra paused. At the center ( r \to 0 ), velocity couldn’t be infinite (no whirlpool tears a hole in reality). So ( C = 0 ). The true function was clean and smooth: Integral calculus including differential equations

Lyra, a young apprentice, faced her final trial: to tame the , a rogue whirlpool deep beneath the city that pulsed with erratic, destructive energy. If she failed, Aethelburg would be torn apart by the year's first monsoon. [ \frac{dv}{dr} + \frac{1}{r} v = 3r^2 ] Lyra paused

[ r v = \int 3r^3 , dr = \frac{3}{4} r^4 + C ] a young apprentice

[ \int_{0}^{4} \frac{3}{4} r^3 , dr = \frac{3}{4} \cdot \left[ \frac{r^4}{4} \right]_{0}^{4} = \frac{3}{16} \left( 4^4 - 0 \right) ]