I have a differential equation that I can't figure out how to solve. It is a first order non-linear ordinary differential equation.
Here it is: v'(t) + R(t)\cdot v^{2/3}=J(t)
I want to solve for v.
It looks like a linear one, but then you see the 2/3. Is there a way to solve this? I know how to integrate R and J if that helps.
$ R(t) = -3^{2/3}$ and $ J(t) = Qt+C_1 \textrm{, where }Q\textrm{ is a constant and }C_1\textrm{ is a constant of integration.}$
This equation can also be written as u'(t) - 1 = J(t)\cdot u^{-2} or w'(t)-2w^{1/2}=2J(t)\cdot w^{-1/2}
using change of variables where $v = \frac{u^3}{3}$, and $w=u^2$.