Question

Find a solution to dA/dt=-6A if A(0)=6. A(t)=

Answer 1

This equation is separable,

$\dfrac{dA}{dt} = -6A \implies \dfrac{dA}A = -6\,dt$

Integrate both sides and solve for $A$ :

$\displaystyle \int \frac{dA}A = -6 \int dt$

$\ln|A| = -6t + C$

$e^{\ln|A|} + e^{-6t+C}$

$\implies A(t) = Ce^{-6t}$

Solve for $C$ using the initial value.

$A(0) = 6 \implies 6 = Ce^0 \implies C=6$

Then the particular solution is

$\boxed{A(t) = 6e^{-6t}}$