Question

find the general form of the function that satisfies \\( \frac{da}{dt}=9a \\).
\\( a(t)= \\)

Explanation:

Step1: Separate variables

We have the differential equation $\frac{dA}{dt}=9A$. Separate variables: $\frac{dA}{A}=9dt$.

Step2: Integrate both sides

Integrate $\int\frac{dA}{A}=\int9dt$. Using the integral formulas $\int\frac{1}{x}dx=\ln|x| + C$ and $\int kdx=kx + C$ (where $k = 9$ is a constant), we get $\ln|A|=9t + C_1$.

Step3: Solve for \(A\)

Exponentiate both sides: $A = e^{9t + C_1}$. Since $e^{9t + C_1}=e^{C_1}\cdot e^{9t}$, and we can let $C = e^{C_1}$ (where $C
eq0$ when $A
eq0$; if $A = 0$, it also satisfies the original differential equation and can be included in the form $A(t)=Ce^{9t}$ with $C = 0$), the general solution is $A(t)=Ce^{9t}$.

Answer:

$A(t)=Ce^{9t}$