Separation of Variables Question 1

If $\dfrac{dy}{dt}=ky$ and $k$ is a nonzero constant, then $y$ could be

$2e^{kty} $

$ 2e^{kt} $

$e^{kt}+3$

$\dfrac{1}{2}ky^2+\dfrac{1}{2}$

Approach

$$\frac{dy}{dt}=ky$$

Separate variables and integrate:

$$ \frac{1}{y} dy = k dt $$ $$ \int \frac{1}{y} dy = \int k dt $$ $$ \ln|y|=kt+C $$ $$ e^{\ln|y|} = e^{kt+C} $$ $$ y=Ae^{kt} $$