Calculus Including Differential Equations | Integral

[ \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) ]

Lyra recognized the form. It was a first-order linear ODE. She rewrote it: Integral calculus including differential equations

[ \frac{dv}{dr} + \frac{1}{r} v = 3r^2 ] [ \int_{0}^{4} \frac{3}{4} r^3 , dr = \frac{3}{4}

[ \frac{d}{dr}(r v) = 3r^3 ]

[ v(r) = \frac{3}{4} r^3 + \frac{C}{r} ] [ \int_{0}^{4} \frac{3}{4} r^3