[ v(r) = \frac{3}{4} r^3 + \frac{C}{r} ]
Lyra recognized the form. It was a first-order linear ODE. She rewrote it: Integral calculus including differential equations
"Here," said her master, old Kael, handing her a data slate. "This equation models how the spin changes with radius. The whirlpool’s total destructive potential is the area under the velocity curve from ( r=0 ) to ( r=R ). Solve for ( v(r) ), then integrate it. That area is the energy you must dissipate." [ v(r) = \frac{3}{4} r^3 + \frac{C}{r} ]
She multiplied through:
Thus, the velocity profile was: