Question

find the general form of the function that satisfies $\frac{da}{dt}=8a$.
$a(t)=square$

Explanation:

Step1: Separate variables

We have the differential equation $\frac{dA}{dt}=8A$. Rearrange it to $\frac{dA}{A}=8dt$.

Step2: Integrate both sides

Integrating $\int\frac{dA}{A}=\int8dt$. The integral of $\frac{1}{A}$ with respect to $A$ is $\ln|A|$, and the integral of $8$ with respect to $t$ is $8t + C$. So, $\ln|A| = 8t+C$.

Step3: Solve for A

Exponentiate both sides: $A = e^{8t + C}=e^{C}e^{8t}$. Let $k = e^{C}$, then $A(t)=ke^{8t}$, where $k$ is an arbitrary constant.

Answer:

$A(t)=ke^{8t}$