How do I use differential equations to get
dM/dt = r(t)M(t)
to equal to
M = E.e^[-(integral from T to t) of r(s)ds]
Where r is the continuous compounded interest which is a function of t and E= M(t) and theT represents some future time.
Where did the ds come from or the r(s) for that fact
The r(s) and the ds both come from being "dummy" variables. What you're doing here to solve the problem is separating the variables: divide both sides by M(t), and then integrate w.r.t. t. The integral on the RHS is equivalent to an indefinite integral w.r.t. t.
Does that make sense?
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