$\frac{dA}{dt} =mgA$ and $A(0)=A$, what is $A(t)$?

Question

$\frac{dA}{dt} = mgA$ where m and g are constant. $A(0)=A_0$ what is $A(t)$?

$$\frac{dA}{dt} = mgA$$ $$\int dA = \int mgA_0 dt$$ $$A(t) = A_0mgt + C$$

This is what I have tried, but according to the answer key, my answer is wrong. The answer should be $A(t) = A_0e^{mgt}$ I'm not sure how to get this.

Answer 1

Hint:

$$\frac{dA}{dt}=mgA$$

$$\frac{dA}{A}=mg\cdot dt$$

Then you may apply integration on both sides to obtain:

$$\ln |A|=mgt+C$$

Answer 2

The problem is that you are wrong in the integral.

$\frac{dA}{dt}=mgA \Rightarrow \frac{dA}{A}=mgdt \Rightarrow$ $\int_{A_{0}}^{A(t)} \frac{dA}{A} $= $\int_{0}^{t} mg dt \Rightarrow \log(A(t))-\log(A_{0})=mgt\Rightarrow \log(\frac{A(t)}{A_{0}})=mgt$ $\Rightarrow \frac{A(t)}{A_{0}}=e^{mgt} \Rightarrow A(t)=A_{0}e^{mgt}$