True and False Question about calculus?

If y changes at a rate proportional to y, then y must be of the form y(t) = Ce^{kt}, with k and C constant. Is the statement true or false?





I think the statement would be false because the growth is proportional so e is not used. But I am not sure if I am right. Thanks for any help.True and False Question about calculus?
It should be true. The statement that y changes at a rate proportional to y can be written as y' is proportional to y


and we make it an equation by using constant of proportionality k, that is y' = ky.





Of course y'= dy/dt so dy/dt = ky.


That is a differential equation which can be solved by


separation of variables. Multiply both sides by 1/y to obtain


1/y (dy/dt) = k


now multiply both sides by dt to effect the separation of variables so we obtain


[1/y]dy = k dt


NEXT integrate both sides with respect to their variable of integration


⌠[1/y]dy = ⌠ k dt


which gives us


ln y + c = kt


Since c is an arbitrary constant, we may name it as Ln C1


since that can be an arbitrary constant too.


now


ln y + ln C1 = ln (C1)y = kt


consequently (C1)y = e^(kt)


so y = [1/C1]e^(kt)


note, however that we may , since C is arbitrary, we may simply make this C=1/C1


so y = C e^(kt)