How does dy/dt=ry end up with the constant multiple on the right?

[oneamp]

Main Question or Discussion Point

I have, from the textbook:

dy/dt = ry

So I solve and get y = e^(rt)+C

However, the book says that y=y_0 * e^(rt), where y_0 is the answer to the initial condition, y(0).

I don't get it. The + C should be what we use to solve for the initial condition; how does y_0 end up in the same term with the exponential?

Thanks

 

[bigfooted]

integrating dy/y=rdt leads to ln(y)=rt+C

this leads to y=exp(rt+C)
but exp(rt+C)=exp(C)*exp(rt)
and you simply replace the constant exp(C) by a new constant C

EDIT: your mistake was most likely that you didn't include the constant of integration immediately after integrating, but after getting rid of the logarithm.

 

[oneamp]

Yes! I see. Thank you.