[math]m_{t+h}-m_{t}=hm_t[/math],

 

so [math]\frac{dm_t}{dt}=m_t[/math]

 

I am told that you use differential equations to the get the differentiation above, but I don't see how. Can anybody see?

Does any of it mean anything? What type of set is m?

Y'what? The first equation is not internally consistant.

 

[math]m_1=2m_0[/math]

 

So far so good. Now derive [math]m_2[/math] directly from [math]m_0[/math], whatever it may be.

 

[math]m_2=3m_0[/math]

 

But deriving it from [math]m_1[/math] above, we run into a contradiction.

 

[math]m_2=2m_1=4m_0[/math]

 

The second equation implies that [math]m_t=e^t+a[/math].

It really should go something like this

 

Let m(t) be a function of time and let h be a small increment of time

 

m(t+h) - m(t) = hm(t)

 

divide through by h

 

(m(t+h)-m(t))/h = m(t)

 

Take limit of LHS as h tends to zero ie h becomes a infinitesimal change in time dt and m(t+dt) - m(t) = dm(t)

ie

dm(t)/dt = m(t)

 

I suspect this is some sort of Markov process