So, I took classes on that 10 years ago and I can't remember anything. Does anyone know how I'd go about solving this?
dk(t)/dt = k(t)/(2t)
given the initial condition k(t_i) = m?
dk(t)/dt = k(t)/(2t)
given the initial condition k(t_i) = m?
I found a differential equations book with the pirates... Turns out this is one of the simplest forms. You can just integrate on both sides:
dk/dt = k/2t
dk/k = dt/2t
\int 1/k dk = 1/2 \int 1/t dt
ln k = 1/2 ln t + c
exp(ln k) = exp(1/2 ln t + c) = exp(1/2 ln t) exp(c) = C exp(1/2 ln t)
k(t) = C t^(1/2)
If k(t_i) = m then
m = C t_i^(1/2)
C = m/t_i^(1/2)
and finally
k(t) = m (t/t_i)^(1/2)
dk/dt = k/2t
dk/k = dt/2t
\int 1/k dk = 1/2 \int 1/t dt
ln k = 1/2 ln t + c
exp(ln k) = exp(1/2 ln t + c) = exp(1/2 ln t) exp(c) = C exp(1/2 ln t)
k(t) = C t^(1/2)
If k(t_i) = m then
m = C t_i^(1/2)
C = m/t_i^(1/2)
and finally
k(t) = m (t/t_i)^(1/2)
man, im really hoping i dont have to learn this stuff
It really sucks that I used to know all that stuff and now it's gone :/
i hate when that happens as well