I can't remember Diff EQ to save my life.

but I have a'(t)=a(t)*a'(0) where a'(0) is constant

my book does

a'(t)/a(t) = d/dt *ln[a(t)] = a'(0)

then they integrate both sides from T to 0 and get

ln[a(t)] = t*a'(0)

we know that if t = 1 ten a(1) = 1 + i, and i some interest rate

then we know ln (1+i) = a'(0)

so basically you get a(t)=(1+i)^t

but I thought from Diff Eq any function of the forum

y'=c*y would give you y =Ce^xt

so how come this one doesn't?